International Journal of Mathematical Education. Volume 1, Number 1 (2011), pp. 63-70 © Research India Publications http://www.ripublication.com/ijme.htm

A Type of Semi-Symmetric Metric Connection in a Lorentzian Para-Sasakian Manifolds

Dhruwa Narain and Sunil Yadav

Department of Mathematics & Statistics D.D.U. Gorakhpur University, Gorakhpur, India

E-mail: profdndubey@yahoo.co.in, prof_sky16@yahoo.com

Abstract

The aim of the present paper is to find the curvature tensor of an LP-Sasakian manifold with respect to connection∇ . And obtained the necessary condition for the projective curvature tensor with respect to connection∇ is coincide to projective curvature tensor of manifold and also proved that if the Ricci tensor of the connection∇ vanishes then the manifold reduces to an −η Einstein manifold.

Mathematics Subject Classification :)2000( 2553,1553 CB Keywords: Semi-Symmetric Metric Connection, an LP-Sasakian manifold, Weyl-conformal curvature tensor, Projective curvature tensor and an −ηEinstein manifold.

Introduction In 1989,K. Matsumoto [2] introduced the notion of an LP-Sasakian manifold. I.Mihai and R.Rosca [4] define the same notion independently after that many authors [3] studied an LP-Sasakian manifold Let ),( gM n be an n –dimensional Riemannian manifold with metric tensor g and∇ be the the Levi-Civita connection on ),( gM n A linear connection∇ on ),( gM n is called semi-symmetric [1] if the torsion tensor T of ∇ has the form (1.1) YXXYYXT )()(),( ππ −= where π is a 1-form on ),( gM n with ρ as associated vector field that is ),()( ρπ XgX = A semi-symmetric connection is called metric connection if it satisfies [5]

64 Dhruwa Narain and Sunil Yadav

(1.2) 0=∇ gX In [1] U.C.De and J.Sengupta define a semi-symmetric metric connection in an almost contact manifold by identifying the 1-form π of )1,1( type with almost contact form η that is given by

(1.3) YXXYYXT )()(),( ηη −= and studied a semi-symmetric metric connection in a locally decomposable Riemannian space whose torsion tensor T satisfies the condition

(1.4) )),(()(),()(),)(( ZYTXAZYTXAZYTx φφ+=∇ In [5] the relation between the semi-symmetric metric connection∇ and the Livi-Civita connection ∇ of ),( gM n is given by

(1.5) ρπ ),()( YXgXYYY XX −+∇=∇ Further, the relation between the curvature tensor R and R of type )3,1( of ∇ and∇ is given by [5] (1.6) LYZXgLXXYgXZYYZXZYXRZYXR ),(),(),(),(),(),( +−−+= αα where

(1.7) ),)((21)()())((),(),( ZYZYZZLYgZY Y ρππππα +−∇==

The Weyl-Conformal curvature tensor in an LP-Sasakian manifold is defined as QYZXgQXZYgYZXXZYZYXRZYXC ),(),(),(),(),(),( −+−+= λλ where

),()2)(1(2

),(1

1),(),( ZYgnn

rZYS

nZQYgZY

−−+

−−==λ ,

S and r is denote the )2,0( Ricci tensor and the scalar curvature of the manifold

),( gM n . In this paper we study a semi-symmetric metric connection on an LP-Sasakian manifold satisfying the condition(1.3) and (1.4).In section 2, the brief idea of an LP-Sasakian manifold are given. In section 3, we find the curvature tensor of semi-symmetric metric connection and prove that if the Ricci tensor of the connection vanishes, then the manifold becomes an −η Einstein manifold. Finally we studied the properties of specific curvature tensor with respect to connection∇ .

A Type of Semi-Symmetric Metric Connection 65

Preliminaries An n -dimension differential manifold ( , )nM g , is called an LP-Sasakian manifolds [3] if it admits a (1,1) − tensor fields φ , a contravariant vector fields ξ , a covariant vector fields η , and Lorentzian metric g of type (0,2) which satisfy

(2.1) ( ) 1η ξ = −

(2.2) 2 ( ) , , , ( ) 1X X X rank nφ η ξ ηοϕ ο ϕξ ο ϕ= + = = = −

(2.3) ( , ) ( , ) ( ) ( )g X Y g X Y X Yφ φ η η= +

(2.4) ( , ) ( ) , Xg X X Xξ η ξ φ= ∇ =

(2.5) ( , ) ( , ) ( , ) ( , )X Y g X Y g X Y Y Xφ φΦ = = = Φ

(2.6 ) ( )( , ) ( , ( ) ) ( )( , )X X XY Z g Y Z Z Y∇ Φ = ∇ Φ = ∇ Φ where, ∇ is the covariant differentiation with respect to Lorentzian metric g the Lorentzian metric g makes a time like unite vector field ,that is ( , ) 1g ξ ξ = − . The manifolds equipped with a Lorentzian almost Para-contact structure ( , , , )gϕ ξ η is said to be Lorentzian almost Para- Contact manifolds ([3],[2]) If we replace ξ by ξ− in (1.1) and (1.2) then the triple ( , , )φ ξ η is an almost Para -Contact structure on defend by Sato [7]. Lorentzian metric given by (1.4) stands analogous to the almost Para contact Riemannian manifolds ([7],[8]).A Lorentzian almost Para -Contact manifold ( , )nM g with the ( , , , )gφ ξ η structure is called a Lorentzian Para-Contact manifold [3] if. 1( , ) {( ) ( ) }2 X YX Y Y Xη ηΦ = ∇ + ∇ A Lorentzian almost Para -Contact manifolds ( , )nM g with these structure ( , , , )gφ ξ ηis called an LP- Sasakian manifold [3] if (2.7) 2( )( ) ( , ) ( )X Y g X Y Y Xφ φ φ η φ∇ = + In an LP-Sasakian manifold the 1-form is closed, further on such an LP-Sasakian manifold ( , )nM g the following relation holds

(2.8) ( ( , ) , ) ( ( , ) ) ( , ) ( ) ( , ) ( )g R X Y Z R X Y Z g Y Z X g X Z Yξ η η η= = −

(2.9) ( , ) ( , ) ( )R X Y g X Y Y Xξ ξ η= −

(2.10) ( , ) ( ) ( )R X Y Y X X Yξ η η= −

(2.11) 2( , ) ( )R X X X Xξ ξ η ξ φ= + =

(2.12) ( , ) ( 1) ( )S X n Xξ η= −

(2.13) ( , ) ( , ) ( 1) ( ) ( )S X Y S X Y n X Yφ φ η η= + −

66 Dhruwa Narain and Sunil Yadav

for any vector fields ,X Y ( , )nM gχ∈ ([3],[9]),where S is the Ricci tensor and Q is the Ricci operator given by ),(),( YQXgYXS = An LP-Sasakian manifold ( , )nM g is said to be η -Einstein if its Ricci tensor S is of the form (2.14) )()(),(),( YXYXgYXS ηβηα += , for any vector field ,X Y ,where βα , are function on ( , )nM g . Curvature tensor of an LP-Sasakian manifold with respect to Semi- symmetric metric Connection Theorem (3.1): The scalar curvature r of an LP-Sasakian manifolds with respect to connection∇ is given by )()1(2)1(3 ξAnnnrr −−−+= Proof: In view of equation (1.3), we get (3.1) YXXYYXT )()(),( ηη −= Contracting (3.1) with respect to Y , we get (3.2) )()1()(),( ZnZTC η−= and (3.3) )()()1()(),( ZnZTC XX η∇−=∇ In view of equation (1.4), we get (3.4) )()()()()1())(,( ZXaAZXAnZTCX ηφη +−=∇ where ))(,( YCa φ= From equation (3.3) and (3.4), we get

(3.5) )()()()())(( ZXAbZXAZX ηφηη +=∇ , where 1−

=n

ab

From equation (2.1) and (2.4), we get (3.6) ),()()()()( ZXgZXXX −−∇=∇ ηηηη Again from equation (3.5) and (3.6), we have (3.7) ),()()())(( ZXgZXZX +=∇ ηηη + )()()()( ZXAbZXA ηφη +

A Type of Semi-Symmetric Metric Connection 67

From equation (1.7) and (3.7), we get

(3.8) ),(23)()()()(),( ZXgZXbAZXAZX −+= ηφηα

Also from (1.7) and (3.8), we have

(3.9) XXbAXALX23)()( −+= ξφξ

Using (3.8)(3.9) in (1.6), we get

(3.10) { } { }{ }{ }{ }ξηφφ

ξηφ),()()()(

),()()()(_),(),(3),(),(ZXgXZYbAYA

ZYgYZXbAXAYZXgXZYgZYXRZYXR

−+−−++−+=

On satiable contracting the equation (3.10) ,we get

(3.11) [ ] ),()()()()()()2(),()1(3),(),( ZYgAZYbAZYAnZYgnZYSZYS ξηφη −+−−−+= And (3.12) )()1(2)1(3 ξAnnnrr −−−+= This complete the prove of the theorem (3.1) Theorem (3.2). If the Ricci tensor of an LP-Sasakian manifold with respected to semi-symmetric metric connection vanishes, Then the manifold becomes anη –Einstein manifold. Proof. From (3.11) it is clear that S is symmetric if and only if (3.13) [ ] [ ] )()()()()()( YZbAZAZYbAYA ηηφ +=+ In particular if 0=S is then from (3.11), we get (3.14) [ ] ),()()()()()()2(),()1(3),( ZYgAZYbAZYAnZYgnZYS ξηφη ++−+−−= Since 0=S this implies that 0=r from (3.12) ,we get

(3.15) ⎥⎦⎤

⎢⎣⎡ +

−= n

n

rA 3

121)(ξ

In view of S is symmetric then from (3.14),we have [ ])()()()( ZYbAZYA ηφη + = [ ] )()()( YZbAZA ηφ+ Replacing ξ=Z in the above equation, we get

(3.16) [ ])()()()( YAbYAYA φηξ +−=

68 Dhruwa Narain and Sunil Yadav

In view of equation (3.14),(3.15) and (3.16) , we get )()(),(),( YXYXgYXS ηβηα +=

where ⎭⎬⎫

⎩⎨⎧ −−

−= )2(3

121

nn

rα and ( )⎭⎬⎫

⎩⎨⎧ −+

−−

−= nnrn

n 3312

21 2β

This complete the prove of the theorem (3.2) Theorem (3.3): The necessary and the sufficient condition for the Ricci tensor of an LP-Sasakian manifold with respect to semi-symmetric metric connection to be symmetric is that 0),(),(),( =++ YXZRXZYRZYXR Proof: Necessary: In view of equation (3.10), we get

{ }{ }{ }{ } { }{ }{ }{ } { }{ }{ }{ }ξηφ

ξηφξηφξηφξηφ

ξηφ

),()()()(),()()()(),()()()(),()()()(),()()()(

),()()()(),(),(),(

YZgZYXbAXA

YXgXYZbAZAXYgYXZbAZA

XZgZXYbAYAZXgXZYbAYA

ZYgYZXbAXAYXZRXZYRZYXR

−+−−++−+−−++−+−

−+=++

On simplification, we get

(3.17) ( )( )( )( ) ( )( )YXXYZbAZAXZZXYbAYA

XYYZXbAXAYXZRXZYRZYXR

)()()()())()()()()()()()(),(),(),(

ηηφηηφηηφ

−++−++−+=++

In view of equation (3.16) from (3.17), we get (3.18) 0),(),(),( =++ YXZRXZYRZYXR Conversely: we suppose that the equation (3.18) holds the from (3.17),we get (3.19) ( ) ( ))()()()()()( YbAYAZZbAZAY φηφη +=+ Putting ξ=Y in the above equation, we get

[ ])()()()( ZAbZAZA φηξ +−= This implies that S is symmetric This complete the prove of the theorem (3.3) Weyl-conformal curvature tensor Theorem (4.1): If the Ricci tensor S of an LP-Sasakian manifold with respect to connection∇ vanishes then weyl-conformal curvature tensor coincide the curvature tensor of the manifold with respect to connection∇ .

A Type of Semi-Symmetric Metric Connection 69

Proof: The weyl-conformal curvature tensor C of type )3,1( of an LP-Sasakian manifold with respect to semi-symmetric connection is defined as

(4.1) YQZXgXQZYgYZXXZYZYXRZYXC ),(),(),(),(),(),( −+−+= λλ where

),()2)(1(2

),(1

1),(),( ZYgnn

rZYS

nZQYgZY

−−+

−−==λ

In particular 0=S ,then 0=r ,therefore above equation reduces as

(4.2) 0),( =ZYλ In view of equation (4.1) and (4.2),we get

(4.3) ZYXRZYXC ),(),( = This complete the prove of the theorem (4.1) Projective curvature tensor Theorem (5.1): If the Ricci tensor of an LP-Sasakian manifold is symmetric, then the necessary condition for the projective curvature tensor with respect to connection∇ is coincide to the projective curvature tensor of the manifold if 0)( =ξA . Proof: The projective curvature tensor P of type )3,1( of an LP-Sasakian manifold with respect to semi-symmetric connection is defined as

(5.1) [ ]YZXSXZYSn

ZYXRZYXP ),(),(1

1),(),( −−

−=

Using (3.10)(3.11) in (5.1) ,we get

(5.2) [ ] [ ][ ] [ ]ξηξηξηηηξ

ηηηξξ

),()(),()()()()()()(

)()()()(12),(),()(

11),(),(

ZXgYZYgXAYXXYZA

XYYXZAn

nYZXgXZYgA

nZYXPZYXP

−+−+

−−−

+−−

+=

Using (2.6) (2.8) in (5.2), we get

(5.3) [ ]ξηξηξξηξ

ξηξξ

),()(),()()(),()()(

),()()(12),()(

11),(),(

ZXgYZYgXAYXRZA

YXRZAn

nZYXRA

nZYXPZYXP

−++−−

+−

+=

From (5.3) it is clear that ZYXPZYXP ),(),( = if 0)( =ξA This complete the prove of the theorem (5.1)

70 Dhruwa Narain and Sunil Yadav

References

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