d.narain and sunil yadav, semi symmetric metric connection in lp-sasaskian manifolds
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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
[1] De, U.C. and Sengupta.J, On a type of semi-symmetric metric connection on an almost contact metric manifold, FACTA UNIVERSITATLS (NIS) SER, MATH. INFORM. 16(2001), 87-96.
[2] Matsumoto K., On Lorentzian Para-contact manifolds, Bull. Of Yamagata Univ. Nat. Sci. 12(2)(1989), 151-156.
[3] Matsumoto K. and Mihai, I.; On a certain transformation in a Lorentzian para-Sasakian manifold, Tensor, N.S. 47(1988), 189-197.
[4] Mihai, I. And Rosca, R.; On Lorentzian P-Sasakian manifolds, Classical Analysis (Kazimierz Dolny) (1991), 155-169, World Scientific, Singapore, 1992. MR. 93e: 53076.
[5] Yano, K.; One semi-symmetric metric connection, Rev. Roumaine Math. Pure App. 15(1970), 1579-1586. MR 0275321 (43#1078).
[6] D. Blair, Contact manifold in Riemannian geometry, Lecture note in math Springer - Verlag, Berlin-Heidelberg, New-York 509(1976).
[7] I. Sato, On a Structure similar to almost Structure, Tensor N S, 30 (1976) 219-224.
[8] I. Sato, On a Structure similar to almost structure II, Tensor N S., 31 (1977) 199-205.
[9] K. Yano and M. Kon, Structure on manifold, Series in Pure Math.3 World.Sci.1984.
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