1
SEMI-SYMMETRIC NON-METRIC CONNECTION IN AN LSP - SASAKIAN MANIFOLDS B. Prasad and Ramashrai Singh Yadav
Dept of Mathematic, G.S. Rai Girls P.G. College Karnai Ballia (U.P.)
Abstract - In a recent paper De and Sengupta (2000) have studied a semi- symmetric non- metric connection on a Riemannian manifold which generalizes the notion of semi- symmetric non- metric connection introduced by Agashe and Chafle in 1992.
In this paper we have studied the properties of curvature tensors of a semi- symmetric non- metric connection in LSP- Sasakian manifold. An expression for the curvature tensor, the Ricci tensor of the connection have been obtain. Further it is also shown that different curvature tensors such as conformal curvature tensor, Projective curvature tensor.
1 INTRODUCTION
Let us consider an n-dimensional differentiable manifold mn endowed with a tensor field
of the type (1, 1), a vector field
, a 1- form and a lorentzian metric g satisfying2
X X ( ) , ( ) X 1
( , ) ( )
g X X
... (1.1)( , ) ( , ) ( ) ( )
g X Y g X Y X Y
... (1.2)An LP- contact manifold is called a Lorentzian Para- Sasakian manifold if (Matsumoto and Mihai, 1988)
X
( ) X
(
X )( ) Y g X Y ( , ) ( ) Y X 2 ( ) ( ) X Y
... (1.3)An LP- -Sasakian Manifold
M
n is said to be Lorentzian Para- Sasakian (LSP-Sasakian Manifold) if it satisfy [Matsumoto and Mihai, 1988]' ( , ) F X Y { ( , ) g X Y ( ) ( )}, X Y
21
... (1.4) The confromal curvature tensor 'V' of the type (0,4), (n>3) is defined as' ( , , , ) ' ( , , , ) 1 { ( , ) ( , ) ( 2)
V X Y Z W R X Y Z W Ric Y Z g X W
n
( , ) ( , ) ( , ) ( , ) ( , ) ( , )}
Ric X Z g Y W Ric X W g Y Z Ric Y W g X Z
{ ( , ) ( , ) ( , ) ( , )}
( 1)( 2)
r g Y Z g X W g X Z g Y W
n n
... (1.5)The projective curvature tensor 'P' of the type (0,4) (m>2) is defined by
' ( , , , P X Y Z W ) ' ( , , , R X Y Z W )
1 { ( , ) ( , ) ( , ) ( , ) 1 Ric Y Z g X W Ric X Z g Y W
n
... (1.6)A Semi -Symmetric non metric connection
in an almost LP-contact metric manifoldM
ncan be defined as in [De and Sengupta, 2000].It is given by
( ) ( , ) ( , ) ,
X
Y
XY Y X g X Y g X Y A
... (1.7)where
anda
are 1-form associated with the vector field
and A anM
n given by( , ) ( )
g X X
... (1.8)and
g(X,A) = a(X).
…... (1.9)2
Remark: If in particular, we take
a =
i.e.A ,
then (1.7) reduces in following form._
( ) ,
X
Y
XY Y X
... (1.10)Where
g X ( , ) ( ). X
Thus we see that the connection defined by De and Sengupta in 2000 is the generalization of a semi-symmetric non- metric connection which is defined by Agashe and Chafle in 1992.
2. CURVATURE TENSOR OF
M
nWITH RESPECT TO SEMI SYMMETRIC NON- METRIC CONNECTION IN AN LSP- SASAKIAN MANIFOLDThe curvature tensor R of
M
n with respect to the semi- symmetric non- metric connection is giving by [De and Sengupta, 2000]( , ) ( , ) (
Y)( ) ( ) ( ) (
x)( ) R X Y Z R X Y Z Z X Y Z X Z Y
( ) ( ) X Z Y g Y Z ( , )[{
XA ( ) A X g X A ( , )
( , ) {
X( ) ( , ) ( , , ) }]
g X A A X g X g X A
( , )[{
Y( ) ( , ) ( , ) } g X Z A A Y g Y A g Y A A
{
Y ( ) Y g Y ( , ) g Y ( , ) }]. A
... (2.1)Using (1.1)(1.3) and (1.4) in (2.1) we get
( , ) ( ,. ) ( , ) ( , ) ( , ) ( , ) R X Y Z R X Y Z g Y Z X g X Z Y g Y Z X A
( , ) ( , ).
g X Z Y A
--- (2.2)Where
( , ) X A
XA ( ) A X g X A ( , ) g X A A ( , ) .
Contracting (2.2), w. r. to X, we get.
( , ) ( , ) ( , ) ( , ){ ( ) ( )
Ric Y Z Ric Y Z g Y Z g Y Z divA n A a ( )} ( ( , ), ).
a A g Y A Z
... (2.3)From (2,3) , we get
{ ( ) ( ) ( )} ( , ).
RY RY Y Y divA n A a a A Y A
... (2.4) Againg contracting (2.4), we get( 1){ ( ) ( ) ( )}.
r r n divA n A a a A
... (2.5)Remark (1) : If in particular, we take
a
i.e. A
, then from (2.2),(2.3) and (2.5), we get( , ) ( , ) ( , ) ( , ) , R X Y Z R X Y Z g X Z Y g Y Z X
( , ) ( , ) ( 1) ( , ), Ric Y Z Ric Y Z n g Y Z
( 1).
r r n n
This is the same expression for the curvature tensor
R
, Ricci tensorRic
and the scalar curvaturer
as were obtained by Jaiswal, Ojha and Prasad in their paper in 2001.From (2.2), we get
' ( , , , R X Y Z W ) ' ( , , , R X Y Z W ) g Y Z g ( , , ) ( X W , ) ( , ) ( , ) ( , ) ( ( , ), ) g X Z g Y W g Y Z g X A W
( , ) ( ( , ), ).
g X Z g Y A W
... (2.6)3 Where
' ( , , , R X Y Z W ) g R X Y Z W ( ( , ) , )
' ( , , , R X Y Z W ) g R X Y Z W ( ( , ) , )
Theorem (2.1): If the LSP-Sasakian manifold
M
nadmits a semi-symmetric non-metric connection, we have.' ( , , , R X Y Z W ) ' ( , , R Y Z X W , ) ' ( , R Z X Y W , , ) 0
... (2.7)( , , , ) ( , , , ) 0
R X Y Z W R Y X Z W
... (2.8)' ( , , , R X Y Z W ) ' ( , , R X Y W Z , ) g X Z g Y W ( , )[ ( , ) g ( ( , ), Y A W )]
( , )[ ( , ) ( ( , ), )]
g X W g Y Z g Y A Z
( , )[ ( , ) ( ( , ), )]
g Y Z g X W g X A W
( , )[ ( , ) ( ( , ), )].
g Y W g X Z g X A Z
... (2.9) In particularly,' ( , , , R X Y Z W ) R X Y W Z ( , , , ) 0,
iff
{ ( g Y W , ) g ( ( , ) )} 0 Y A W
' ( , , , R X Y Z W ) ' ( , R Z W X Y , , ) g X Z g ( , )[ ( ( , , ), W A Y ( ( , , ), )] ( , )[ ( , )
g Y A W g Y Z g X W
( ( , , ), )] ( , )[ ( , ) g X A W g X W g Z Y
( ( , , ), )]
g Z A Y
... (2.10)Proof: Taking cyclic sum of (2.6) and using Bianchi's first identity we get (2.7). Similarly other results can also be proved.
Remark 2: If we take
a
i.e.A ,
then from (2, 9)R X Y Z W ( , , , )
+' ( , , R X Y W Z , ) 0
and (2.10)' ( , , R X Y W Z , ) ' ( , R Z W X Y , , ) 0.
This is the same expression as was proved by Jaiswal, Ojha and Prasad in their paper in 2001.3. SYMMETRIC CONDITION OF THE RICCI TENSOR
:Theorem (3.1): If the LSP-Sasakian manifold (
M
n) admits a semi symmetric non-metric connection
, then a necessary and sufficient condition for the Ricci tensor of
to besymmetry is that.
( ( , ), ) ( ( , ), ).
g Y A Z g Z A Y
Proof: By virtue of (2.3), we get
( , ) ( , ) ( , ) ( , ){ ( ) ( )
Ric Z Y Riic Z Y g Z Y g Z Y divA n A a ( )} ( ( , ), ).
a A g Z A Y
... (3.1)From (2.3) and (3.1), we get
( , ) ( , ) ( ( , ), ) ( ( , ), ).
Ric Y Z Ric Z Y g Z A Y g Y A Z
... (3.2) If
Ric X Y ( , )
is symmetric the left hand side of (3.2) vanishes and we get.( ( , , ), ) ( ( , ), ).
g Y A Z g Z A Y
... (3.3)4 Hence proves the theorem.
4. SKEW-SYMMETRIC CONDITION OF THE RICCI TENSOR
:Theorem (4.1): If the LSP- Sasakian manifod
( M
n)
admits a semi-symmetric non-metric connection
, then a necessary and sufficient condition for the Ricci tensors of
to beskew-symmetric is that the Ricci tensor of the Riemannian connection
is given by.( , ) ( , ) ( , ){ ( ) ( ) ( )}
Ric Y Z g Y Z g Y Z divA n A a a A 1 { ( ( , , ) ) ( ( , ), )}
2 g Y A Z g Z A Y
Proof: From (2.3) and (3.1), we get
( , ) ( , ) 2 ( , ) 2 ( ) 2 ( , ){
Ric Y Z Ric Z Y Ric Y Z g YZ g Y Z divA ( ) ( ) ( )} ( ( , ),
n A a a A g Y A Z
( ( , ), ).
g Z A Y
... (4.1)If
Ric X Y ( , )
is skew-symmetric the left side of (4.1) vanishes and we get( , ) ( , ) ( , ){ ( ) ( ) ( )}
Ric Y Z g Y Z g Y Z divA n A a a A 1 { ( ( , ), ) ( ( , ), )}
2 g Y A Z g Z A Y
... (4.2)On the other hand, if Ric (Y, Z) is given by (4.2) then from (4.1), we get
( , ) ( , ) 0 Ric Y Z Ric Z Y
Hence proves the theorem.
Also, if Ric (Y,Z)=0 , then from (4.2), we get
1 { ( ( , ), ) ( ( , ), )} ( , ) ( , ) 2
{ ( ) ( ) ( )}
g Y Z Z g Z Z Y g Y Z g Y Z
divA n A a a A
Which is not possible? Hence we can write.
Corollary: If the LSP- Sasakian manifold
( M
n)
admits a semi-symmetric non-metric connection whose Ricci-tensor is skew- symmetric, then the manifold cannot be Ricci flat.5. CONFORMAL CURVATURE TENSOR OF SEMI SYMMETRIC NON-METRIC CONNECTION IN AN LSP- SASAKIAN MANIFOLD
Theorem (5.1): In an LSP-Sasakian manifold the conformal curvature tensor of the semi- symmetric non- metric connection and the Riemannian connection are not equal i.e.
' ( , , , V X Y Z W ) ' ( , , , V X Y Z W ).
Proof: Analogous to the definition (1.5), we define conformal curvature tensor of
M
n with respect to the semi-symmetric non-metric connection in an LSP -Sasakian manifold by' ( , , , ) ' ( , , , ) 1 { ( , ) ( , ) V X Y Z W R X Y Z W 2 Ric Y Z g X W
n
( , ) ( , ) ( , ) ( , ) Ric X Z g Y W g X Z Ric Y W
( , ) ( , ) { ( , ) ( , )
( 1)( 2)
g Y Z Ric X W r g Y Z g X W
n n
( , ) ( , } g X Z g Y W
... (5.1)5
In consequences of (1.5), (2.3),(2.5), (2.6) and (5.1), we get
1
' ( , , , ) ' ( , , , ) [ ( , ){ ( , ) 2
V X Y Z W V X Y Z W n g X Z g Y W
n
( ( , ), )} ( , ){ ( , ) g Y A W g Y Z g Y W
( ( , ), )}] 1 [ ( , ) ( , ) g X A W 2 g Y W g X Z
n
( ( , ), )} ( , ){ ( , ) g X A Z g X W g Y Z
( ( , ), )}] 1 [ ( , ) ( , ) g Y A Z 2 g Y W g X Z
n
, ) ( , )][ ( ) ( ) ( )].
gY Z g X W divA n A a a A
... (5.2) Hence prove the theorem.
Theorem (5.2): The conformal curvature tensor
' ( , , , V X Y Z W )
of the semi-symmetric non-metric connection in an LSP -Sasakian manifold satisfies the following algebraic properties.
' ( , , , V X Y Z W ) ' ( , V Y X Z W , , ) 0
... (5.3)( , ) ( , ) ( , ) 1 [{ ( ( , ), ) V X Y Z V Y Z X V Z X Y 2 g Y A Z
n
g ( ( , ) )} Z A Y X { ( ( , ), g Z A X )
g ( ( , ) )} X A Z Y { ( ( , ), ) g X A Y
g ( ( , ), Y A X )} ]. Z
... (5.4) In particular( , ) ( , ) ( , ) 0
V X Y Z V Y Z X V Z X Y
If
{ ( ( , ), g Y A Z g ( ( , ), )} 0 Z A Y
Remark: If
a
i.eA
, then from (5.2), we get' ( , , , V X Y Z W ) ' ( , , , V X Y Z W ).
Hence, in an LSP-Sasakian manifold the conformal curvature tensors of semi- symmetric non-metric connection and Riemannian connection are equal. This theorem was obtained by Jaiswal, Ojha and Prasad in their paper in 2001.
6. PROJECTIVE CURVATURE TENSOR OF SEMI SYMMETRIC NON-METRIC CONNECTION IN AN LSP -SASAKIAN MANIFOLD
Theorem (6.1): In an LSP-Sasakian manifold the projective curvature tensor of the semi- symetric non-metric connection and the Riemannian connection are not equal.
i.e.
P X Y Z ( , , ) P X Y Z ( , ) .
Proof: Analogous to the definition (1.6) we define projective curvature tensors of
M
nwith respect to the semi-symmetric non-metric connection in an LSP- Sasakian manifold by( , ) ( , , ) 1 [ ( , ) ( , ) ].
P X Y Z R X Y Z 1 Ric Y Z X Ric X Z Y
n
... (6.1) By using (2.2), (2.3),(1.6)and (6.1), we get
P X Y Z ( , ) P X Y Z ( , ) [ ( , ){ g Y Z X ( , )} X A g X Z ( , ){ Y
6
) , )}] 1 [{ ( , ) ( ( , ), ) Y A 1 g Y Z g Y A Z
n
( , )( ( ) ( ) ( )} {( , ) g Y Z divA n A a a A X X Z
( ( , ) ) ( , )( ( ) ( ) g X A Z g X Z divA n A a
( ) }].
a A Y
... (6.2)Hence prove the theorem.
Theorem (6.2): The projective curvature tensor
P X Y Z ( , )
of the semi symmetric non- metric connection in an LSP-Sasakian manifold satisfies the following algebraic properties( , ) ( , ) 0 P X Y Z P Y X Z
( , , ) ( , ) ( , ) 1 [{ ( ( , ), ) P X Y Z P Y Z X P Z X Y 1 g Y A Z
n
( ( , ), )} { ( ( , ), ) g Z A Y X g Z A X
( ( , ) )} { ( ( , ), ) g X A Z Y g X A Y
( ( , ), )} ].
g Y A X Z
... (6.3)REFERENCES
1. Chaki, M.C, and Trafdar, M. (1988): On conformally flat pseudo Ricci-Symmetric manifold, period Math.
Hung. Vol 19, PP.209-215.
2. Chaki, M.C, (1988) on pseudo Ricci, symmetric manifolds, Bulg. J. Phys, 15, 526-530.
3. De, U.C. and Sengupta, J. (2000): On a type of semi-symmetric non- metric connection, Bull cal. Math.
Soc. 92(5), 375-384.
4. Adati, T. and Matsumoto, K. (1977) : On conformally, recurrent and conformally symmetric P- sasakiam manifolds, T.R.U, Maths. Vol. 13, 25-32
5. Nirmala, S Agashe and Mangla, R. Chafle (1992); A semi-symmetric non- metric connection on a Riemannian manifold. Indian J.Pure App, Math, 23(6), 399-409.
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