Vol 3, No 1 (2012)

(1)

Available online through

www.ijma.info

QUARTER-SYMMETRIC METRIC CONNECTION ON A SASAKIAN MANIFOLD

Riddhi Jung Shah*

Department of Mathematics & Astronomy, Lucknow University, Lucknow – 226007, India

E-mail: [email protected]

(Received on: 05-01-12; Accepted on: 26-01-12)

________________________________________________________________________________________________

ABSTRACT

The purpose of the present paper is to study a quarter-symmetric metric connection on a Sasakian manifold. We study

the concircular curvature tensor with respect to the quarter-symmetric metric connection satisfying the

condition

R

~

(

ξ

,

U

).

Z

~

=

0

. We also study

ξ

-pseudo projectively flat and pseudo-projectively flat Sasakian manifold with respect to the quarter-symmetric metric connection. Next, we investigate the nature of semi-symmetric Sasakian manifold admitting quarter-symmetric metric connection. Finally we study quasi-conformally flat Sasakian manifold with respect to the quarter-symmetric metric connection and prove that the manifold is an

η

-Einstein manifold.

Keywords: Quarter-symmetric metric connection, concircular curvature tensor, pseudo-projective curvature tensor, quasi-conformal curvature tensor, Sasakian manifold.

Mathematics Subject Classification 2010: 53B15, 53C15, 53C25.

________________________________________________________________________________________________

1. INTRODUCTION:

In this paper we study the quarter-symmetric metric connection on a Sasakian manifold. In 1975, S. Golab [6] defined and studied quarter-symmetric metric connection in a differentiable manifold with affine connection.

A linear connection

∇

~

on an

n

-dimensional Riemannian manifold

(

M

,

g

)

is called a quarter-symmetric connection [6] if its torsion tensor

T

~

of the connection

∇

~

]

,

[

~

)

,

(

~

Y

X

Y

X

T

=

∇

_X

−

∇

_Y

−

satisfies

(1.1)

T

~

(

X

,

Y

)

=

η

(

Y

)

ϕ

X

−

η

(

X

)

ϕ

Y

,

where

η

is a 1-form and

ϕ

is a (1,1) tensor field. In particular, if

ϕ

(

X

)

=

X

, then the quarter-symmetric connection reduces to the semi-symmetric connection [5]. Thus the notion of quarter-symmetric connection generalizes the idea of semi-symmetric connection.

A quarter-symmetric connection

∇

~

is said to be a quarter-symmetric metric connection if it satisfies the condition

(1.2)

(

∇

~

_X

g

)(

Y

,

Z

)

=

0 ,

for all

X

,

Y

,

Z

∈

TM

,

where TM is the Lie algebra of vector fields of the manifold M, otherwise

∇

~

is said to be a Quarter-Symmetric non- metric connection.

Quarter-Symmetric metric connection is also studied by S. C. Rastogi [10], R. S. Mishra and S. N. Pandey [7], K. Yano and T. Imai [12], U. C. De and S. C. Biswas [2] etc.

________________________________________________________________________________________________

(2)

& '( $)% $)*

A Sasakian manifold is said to be an

η

-Einstein manifold if its Ricci tensor S of type (0, 2) satisfies the condition

( , )

( ) ( ),

S X Y

=

α

g X Y

+

β η

X

η

Y

where

α

and

β

are smooth functions. In particular, if

β

=0, then the Sasakian manifold is said to be an Einstein manifold.

The paper is organized as follows:

In preliminaries we study some known basic result in a Sasakian manifold and in the quarter-symmetric metric

connection

∇

.

Section 3 deals with the concircular curvature tensor with respect to the quarter-symmetric metric connection on a Sasakian manifold. In this section we prove that if in a Sasakian manifold concircular curvature tensor

with respect to the quarter-symmetric metric connection satisfies the condition

R

~

(

ξ

,

U

).

Z

~

=

0

then the manifold is an Einstein manifold with respect to the quarter-symmetric metric connection. In section 4 we investigate

ξ

-pseudo projectively flat Sasakian manifold with respect to the quarter-symmetric metric connection and prove that the manifold is

ξ

-pseudo projectively flat with respect to the Levi-Civita connection

∇

if and only if it is so with respect to

∇

~

under the condition

a

=

(

1 −

n

)

b

.

Pseudo- projectively flat Sasakian manifold with respect to quarter-symmetric metric connection is studied in section 5. In the next section semi-symmetric Sasakian manifold admitting quarter- symmetric metric connection is studied and it is proved that the manifold is an Einstein manifold. Finally we study the quasi-conformally flat Sasakian manifold with respect to the quarter-symmetric metric connection and it is also proved that the manifold in such a condition is an

η

-Einstein manifold.

2. PRELIMINARIES

An

n

(

=

2 m

+

1 )

dimensional smooth manifold M is said to be a contact manifold if it carries a global 1-form

η

such that

∧

(

)

m

≠

0 d

η

everywhere on M. For a given contact 1-form

η

there exists a unique vector

ξ

such that

0 )

,

(

X

=

d

η

ξ

and

η

(

ξ

)

=

1

. Polarizing

d

η

on the contact subbundle

η

=

0

, one obtains a Riemannian metric g and a (1, 1) tensor field

ϕ

such that

(2.1)

d

η

(

X

,

Y

)

=

g

(

ϕ

X

,

Y

),

η

(

X

)

=

g

(

X

,

ξ

),

ϕ

2

X

=

−

X

+

η

(

X

)

ξ

,

g is called an associated metric of

η

and

(

ϕ

,

ξ

,

η

,

g

)

a contact metric structure. If the contact metric structure of M is normal then M is said to have a Sasakian structure. An almost contact metric structure

(

ϕ

,

ξ

,

η

,

g

)

on M is a Sasakian structure if the relation

(2.2)

(

∇

_X

ϕ

)

Y

=

g

(

X

,

Y

)

ξ

−

η

(

Y

)

X

holds, where

∇

denotes the Levi-Civita connection of g. The manifold M equipped with the Sasakian structure is called a Sasakian manifold. The contact structure on M is said to be normal if the almost complex structure on

M

×

is defined by

=

−

dt

d

X

f

X

dt

d

f

X

J

,

ϕ

ξ

,

η

(

)

,

where f is a real function on

M

×

, is integrable. It is well known that every Sasakian manifold is K-contact but converse is not true in general.

In a Sasakian manifold

(

M

n

,

g

)

the following relations hold [3], [4], [12]:

(2.3)

ϕξ

=

0 ,

η

(

ξ

)

=

1 ,

η

o

ϕ

=

0 ,

(3)

& '( $)% $)*

(2.6)

(

∇

_X

η

)

Y

=

g

(

X

,

ϕ

Y

)

,

(2.7)

R

(

X

,

Y

)

ξ

=

η

(

Y

)

X

−

η

(

X

)

Y

,

(2.8)

R

(

ξ

,

X

)

Y

=

(

∇

_X

ϕ

)

Y

=

g

(

X

,

Y

)

ξ

−

η

(

Y

)

X

,

(2.9)

η

(

R

(

X

,

Y

)

Z

)

=

g

(

Y

,

Z

)

η

(

X

)

−

g

(

X

,

Z

)

η

(

Y

),

(2.10)

S

(

X

,

ξ

)

=

(

n

−

1 )

η

(

X

),

(2.11)

S

(

ϕ

X

,

ϕ

Y

)

=

S

(

X

,

Y

)

−

(

n

−

1 )

η

(

X

)

η

(

Y

)

, for any vector fields X, Y, Z.

A quarter-symmetric metric connection

∇

~

with torsion tensor (1.1) in a Sasakian manifold is given by [8]

(2.12)

∇

~

_X

Y

=

∇

_X

Y

−

η

(

X

)

ϕ

Y

.

Let

R

~

and R be the curvature tensor with respect to the connection

∇

~

and

∇

respectively. Then we have [9]

(2.13)

),

(

}

)

(

)

(

{

)

,

(

)

(

)

,

(

)

(

)

,

(

2 )

,

(

)

,

(

~

Z

Y

X

Y

Z

X

g

Y

Z

Y

g

X

Z

Y

X

d

Z

Y

X

R

Z

Y

X

R

η

ξ

η

ξ

η

ϕ

η

−

+

−

+

−

=

where

R

(

X

,

Y

)

Z

is the Reimannian curvature of the manifold.

Also from (2.13), we obtain

(2.14)

S

~

(

Y

,

Z

)

=

S

(

Y

,

Z

)

−

2 d

η

(

ϕ

Z

,

Y

)

+

g

(

Y

,

Z

)

+

(

n

−

2 )

η

(

Y

)

η

(

Z

),

where

S

~

and

S

are the Ricci tensor of the connection

∇

~

and

∇

respectively. From (2.14) it is clear that

S

~

is symmetric

Again, contracting (2.14), we have

(2.15)

~

r

=

r

+

2 (

n

−

1 )

where

r

~

and

r

are the scalar curvature of the connection

∇

~

and

∇

respectively.

From (2.14) we also have

(2.16)

S

~

(

Y

,

ξ

)

=

2 (

n

−

1 )

η

(

Y

)

.

3. CONCIRCULAR CURVATURE TENSOR SATISFYING

R

~

(

ξ

,

U

).

Z

~

=

0

Let

(

M

,

g

)

be an

n

-dimensional Riemannian manifold. Then the concircular curvature tensor Z is defined by [13]

(3.1)

(

,

)

(

,

)

),

)

1 (

)

,

(

)

,

(

g

Y

U

X

g

X

U

Y

n

r

U

Y

X

R

U

Y

X

Z

−

=

for all

X

,

Y

,

U

∈

TM

, where

r

is the scalar curvature of M. The concircular curvature tensor of a Sasakian manifold with respect to the quarter-symmetric metric connection

∇

~

is defined by

(3.2)

(

,

)

(

,

)

).

)

1 (

~

)

,

(

~

)

,

(

~

Y

U

X

g

X

U

Y

g

n

r

U

Y

X

R

U

Y

X

Z

−

=

Using the equation (2.13) and (2.15) in (3.2) we get

(3.3)

( , )

( ) ( , )

{ ( )

( ) } ( ) 2

( , )

2 ( ( , )

( , ) ),

Z X Y U

X g Y U

Y g X U

Y X

X Y

U

d

X Y U

g Y U X

g X U Y

n

η

ξ η

ξ

η

φ

=

+

−

+

−

(4)

& '( $)% $)*

where Z is the concircular curvature tensor defined by (3.1). Putting

U

=

ξ

in the equation (3.3) and using (3.1) and (2.7), we obtain

(3.4)

(

)

(

)

),

)

1 (

)

1 (

2 )

,

(

~

2

Y

X

Y

n

r

n

Y

X

Z

ξ

η

−

η

−

=

which implies

(3.5)

(

)

),

)

1 (

)

1 (

2 )

,

(

~

2

Y

n

r

n

Y

Z

−

=

η

ξ

and

(3.6)

(

)

.

)

1 (

)

1 (

2 )

,

(

~

2

ξ

η

ξ

X

n

r

n

X

Z

−

=

From (3.4) we also have

(3.7)

η

(

Z

~

(

X

,

Y

)

ξ

)

=

0 .

Let us consider an

n

-dimensional Sasakian manifold admitting quarter-symmetric metric connection

∇

,

which

satisfies the condition

R

~

(

ξ

,

U

).

Z

~

=

0

, then we have

(3.8)

R

~

(

ξ

,

U

).

Z

~

(

X

,

Y

)

ξ

=

0

,

which implies

(3.9)

R

( , ) ( , )

ξ

U Z X Y

ξ

−

Z R

( ( , ) , )

ξ

U X Y

ξ

−

Z X R

( , ( , ) )

ξ

U Y

ξ

−

Z X Y R

( , ) ( , )

ξ

U

ξ

=

0.

From the equation (2.8) and (2.13), we obtain

(3.10)

R

~

(

ξ

,

X

)

U

=

2 {

g

(

X

,

U

)

ξ

−

η

(

U

)

X

}

.

By the use of (3.10) and (3.7) in (3.9) we have

(3.11)

( , ( , ) )

( , ) ( , )

( ) ( , )

( , )

0. g U Z X Y

g X U Z

Y

g Y U Z X

X Z U Y

Y Z X U

U Z X Y

Z X Y U

ξ ξ

ξ

ξ ξ η

ξ η

ξ

η

ξ

−

+

−

+

=

By virtue of (3.4), (3.5) and (3.6) in (3.11), we obtain

(3.12)

{

(

,

)

(

,

)

}

~

(

,

)

0 .

)

1 (

2 )

1 (

2 =

+

−

U

Y

X

Z

X

U

Y

g

Y

U

X

g

n

r

n

Using (3.2) and (2.15) in (3.12) we get

(3.13)

R X Y U

( , )

=

2{ ( , )

g Y U X

−

g X U Y

( , ) }.

Taking inner product on both side of (3.13) by W and contracting over X and W we get

(3.14)

S

~

(

Y

,

U

)

=

2 (

n

−

1 )

g

(

Y

,

U

).

Thus we can state the following theorem:

Theorem 3.1: An

n

-dimensional Sasakian manifold admitting quarter-symmetric metric connection

∇

~

with condition

0 ~

).

,

(

~

=

Z

U

(5)

& '( $)% $)*

4. PSEUDO-PROJECTIVE CURVATURE TENSOR ON A SASAKIAN MANIFOLD:

The pseudo-projective curvature tensor in an

n

-dimensional Riemannin manifold is given by [1]

(4.1)

( , )

[ ( , )

( , ) ]

{ ( , )

( , ) }

1 r

a

P X Y Z

aR X Y Z

b S Y Z X

S X Z Y

b

g Y Z X

g X Z Y

n n

=

+

−

+

−

where

a b

,

are constants such that

a

,

b

≠

0

and

R S r

, ,

are the curvature tensor, Ricci tensor and scalar curvature respectively. The pseudo-projective curvature tensor of a Sasakian manifold with respect to the quarter-symmetric metric connection is defined as

(4.2)

( , )

[ ( , )

( , ) ]

{ ( , )

( , ) },

1 r

a

P X Y Z

aR X Y Z

b S Y Z X

S X Z Y

b

g Y Z X

g X Z Y

n n

=

+

−

+

−

where

R

~

,

S

~

and

~

r

are the curvature tensor, Ricci tensor and Scalar curvature with respect to the quarter-symmetric metric connection

∇

~

respectively.

By the use of equations (2.13), (2.14) and (2.15) in (4.2), we get

(4.3)

( )

( , )

2 ( , )

( ) ( , )

{

(

2) }{ ( )

( ) ( ) }

2 (

, )

2 (

,

)

2{

(

1) }

{ ( , )

( , ) }

P X Y Z

ad

X Y

Z

a

X g Y Z

a Y g X Z

a

n

b

Y

Z X

X

Z Y

bd

Z Y X

bd

Z X Y

a

n

b

g Y Z X

g X Z Y

n

η

φ

η

ξ

η

ξ

η

η φ

=

−

+

−

+

−

+

−

+

−

where

P

is the pseudo-projective curvature tensor defined by (4.1).

Putting

Z

=

ξ

in (4.3), we obtain

(4.4)

~

(

,

)

(

,

)

(

2 )

{

a

(

n

1 )

b

}{

(

Y

)

X

(

X

)

Y

}

n

Y

X

P

Y

X

P

ξ

=

ξ

+

−

+

−

η

−

η

.

If

a

=

(

1 −

n

)

b

,

then (4.4) reduces to

(4.5)

P

~

(

X

,

Y

)

ξ

=

P

(

X

,

Y

)

ξ

.

Again, by the use of (2.7) and (2.10) in (4.1) we obtain

(4.6)

{

(

)

(

)

},

)

1 (

1 }

)

1 (

{

)

,

(

Y

X

Y

n

r

b

n

a

Y

X

P

ξ

η

−

η

−

+

=

which implies

P

(

X

,

Y

)

ξ

=

0

if either

a

=

(

1 −

n

)

b

or

r

=

n

(

n

−

1 )

.

Hence we can state a theorem.

(6)

& '( $)% $)*

5. PSEUDO- PROJECTIVELY FLAT SASAKIAN MANIFOLD ADMITTING THE QUARTER-SYMMETRIC METRIC CONNECTION:

Definition:1 Let M be an

n

-dimensional Sasakian manifold with respect to the quarter-symmetric metric

connection

∇

.

Then the manifold M is said to be pseudo- projectively flat with respect to the quarter-symmetric metric

connection

∇

~

if

(5.1)

P

~

(

X

,

Y

)

Z

=

0 ,

for all

X

,

Y

,

Z

∈

TM

.

Now, from (4.3) and (5.1) we have

(5.2)

( , )

2 ( , )

( ) ( , )

{

(

2) }{ ( ) ( )

( ) ( ) }

2 (

, )

2 (

,

)

2{

(

1) }

{ ( , )

( , ) }.

P X Y Z

ad

X Y

Z

a

X g Y Z

a

Y g X Z

a

n

b

Y

Z X

X

Z Y

bd

Z Y X

bd

Z X Y

a

n

b

g Y Z X

g X Z Y

n

η

φ

η

ξ

η

ξ

η

η φ

=

−

+

−

+

−

+

−

+

−

Using (4.1) in (5.2) we obtain

(5.3)

( , )

[ ( , )

( , ) ]

(

1)

{ ( , )

( , ) }

1

2 ( , )

( ) ( , )

{

(

2) }{ ( ) ( )

( ) ( ) } 2

(

, )

2{

(

1) }

2 (

,

)

{ ( , )

( , ) }.

aR X Y Z

b S Y Z X

S X Z Y

r

a

n

b

g Y Z X

g X Z Y

n

ad

X Y

Z

a

X g Y Z

a Y g X Z

a

n

b

Y

Z X

X

Z Y

bd

Z Y X

a

n

b

bd

Z X Y

b

g Y Z X

g X Z Y

n

η

φ

η

ξ

η

ξ

η

η φ

+

−

+

−

=

−

+

−

+

−

+

−

Taking inner product on both sides of (5.3) by

ξ

and using (2.4) and (2.9), we get

(5.4)

( , ) ( )

2 (

1)

(3

2)

{

(

1) }

{ ( , )

( )

( , )

( )

}.

(

1)

a n

b n

r a

n

b

bS Y Z

X

bS X Z

Y

g Y Z

X

g X Z

Y

n

n n

η

=

η

+

−

− +

−

+

+ −

η

−

η

−

Putting

X

=

ξ

in (5.4) and using (2.10), we get

(5.5)

).

(

)

(

)

1 (

}

)

1 (

{

2

4

2 )

1 (

2

1 )

,

(

)

1 (

}

)

1 (

{

)

1 (

2 )

2

3 (

1 )

,

(

Z

Y

n

b

n

a

r

n

b

nb

b

n

a

b

Z

Y

g

n

b

n

a

r

n

a

n

b

Z

Y

S

η

−

+

−

+

−

+

−

+

−

+

−

=

Let

{

e

_i

:

i

=

1 ,

2 ,...,

n

}

be an orthonormal basis of the tangent space at any point of the manifold. Putting

i

e

Z

Y

=

in (5.5) and taking summation over i,

1 ≤

i

≤

n

, we get

(5.6)

2 (

1 ){

(

1 )

(

2

1 )}

,

−

≠

0 −

−

=

if

a

b

a

n

b

n

a

n

(7)

& '( $)% $)*

In view of (5.5) and (5.6), we obtain

(5.7)

S Y Z

( , )

a

(2

n

3)

b n

(4

3)

g Y Z

( , )

b n

(3

2)

a n

(

2)

( ) ( )

Y

Z

a b

η

−

=

+

−

If

a b

− ≠

0.

From equation (5.7), we have

(5.8)

S

(

Y

,

Z

)

=

Ag

(

Y

,

Z

)

+

B

η

(

Y

)

η

(

Z

)

,

where (5.9)

−

=

b

a

n

b

n

a

A

(

2

3 )

(

4

3 )

and (5.10)

−

=

b

a

n

a

n

b

B

(

3

2 )

(

2 )

.

This leads to the following:

Theorem 5.1: If a Sasakian manifold is pseudo-projectively flat with respect to the quarter-symmetric metric

connection

∇

~

then the manifold is an

η

-Einstein manifold provided that

a

−

b

≠

0

.

6. SEMI SYMMETRIC SASAKIAN MANIFOLD ADMITTING THE QUARTER- SYMMETRIC METRIC CONNECTION:

Definition: 1 A Sasakian manifold

(

M

n

,

g

)

is said to be semi-symmetric if it satisfies the relation [11]

,

0 ).

,

(

X

Y

R

=

R

where

R

(

X

,

Y

)

is the curvature operator.

Let us consider a semi-symmetric Sasakian manifold admitting the quarter-symmetric metric connection

∇

.

Then we have

(6.1)

(

R

~

(

ξ

,

X

).

R

~

)(

U

,

V

)

W

=

0

,

which implies

(6.2)

R

( ,

ξ

X R U V W

) ( , )

−

R R

( ( ,

ξ

X U V W

) , )

−

R U R

( , ( ,

ξ

X V W

) )

−

R U V R

( , ) ( ,

ξ

X W

)

=

0.

Using (3.10) in (6.2), we get

(6.3)

.

0 )

,

(

~

)

(

)

,

(

~

)

,

(

)

,

(

~

)

(

)

,

(

~

)

,

(

)

,

(

~

)

(

)

,

(

~

)

,

(

)

,

(

~

(

)

,

(

~

,

(

=

+

−

+

−

+

−

X

V

U

R

W

V

U

R

W

X

g

W

X

U

R

V

W

U

R

V

X

g

W

V

X

R

U

W

V

R

U

X

g

X

W

V

U

R

W

V

U

R

X

g

η

ξ

η

ξ

η

ξ

η

ξ

Taking inner product on both side of (6.3) by

ξ

we obtain

(8)

& '( $)% $)*

Now, from (2.13) and (3.10) we have

(6.5)

η

(

R

~

(

U

,

V

)

W

)

=

2 η

(

U

)

g

(

V

,

W

)

−

2 η

(

V

)

g

(

U

,

W

),

(6.6)

η

(

R

~

(

ξ

,

V

)

W

)

=

2 g

(

V

,

W

)

−

2 η

(

V

)

η

(

W

),

(6.7)

η

(

R

~

(

U

,

ξ

)

W

)

=

2 η

(

U

)

η

(

W

)

−

2 g

(

V

,

W

),

(6.8)

η

(

R

~

(

U

,

V

)

ξ

)

=

0 .

By Virtue of (6.5), (6.6), (6.7) and (6.8), (6.4), reduces to

(6.9)

g

(

X

,

R

~

(

U

,

V

)

W

)

−

2 g

(

X

,

U

)

g

(

V

,

W

)

+

2 g

(

X

,

V

)

g

(

U

,

W

)

=

0 ,

which implies

(6.10)

g

(

X

,

R

~

(

U

,

V

)

W

)

=

2 [

g

(

V

,

W

)

g

(

X

,

U

)

−

g

(

U

,

W

)

g

(

X

,

V

)]

.

From (6.10), we get

(6.11)

R

~

(

U

,

V

)

W

=

2 [

g

(

V

,

W

)

U

−

g

(

U

,

W

)

V

].

Hence we can state:

Theorem: 6.1 A semi-symmetric Sasakian manifold

(

M

n

,

g

)

admitting quarter-symmetric metric connection

∇

~

is a manifold of constant curvature 2 with respect to the quarter-symmetric metric connection.

Let

{

e

_i

:

i

=

1 ,

2 ,...,

n

}

be an orthonormal basis of the tangent space at any point of the manifold. After taking inner

product with

Z

,

putting

U

=

Z

=

e

_i

in (6.11) and summing over i,

1 ≤

i

≤

n

, we get

(6.12)

S V W

( ,

)

=

2 ( 1) ( ,

n

−

g V W

)

.

This leads to the following result:

Theorem: 6.2 A semi-symmetric Sasakian manifold

(

M

n

, )

g

admitting quarter-symmetric metric connection is an Einstein manifold with respect to the quarter-symmetric metric connection.

7. QUASI- CONFORMALLY FLAT SASAKIAN MANIFOLD WITH RESPECT TO THE QUARTER-SYMMETRIC METRIC CONNECTION:

In a Riemannian manifold

(

M

n

,

g

)

the quasi-conformal curvature tensor W is defined by [14]

(7.1)

},

)

,

(

)

,

(

{

2

1 ]

)

,

(

)

,

(

)

,

(

)

,

(

[

)

,

(

)

,

(

Y

Z

X

g

X

Z

Y

g

b

n

a

n

r

QY

Z

X

g

QX

Z

Y

g

Y

Z

X

S

X

Z

Y

S

b

Z

Y

X

aR

Z

Y

X

W

−

+

−

+

−

+

=

where R, S, Q and r are the curvature tensor, Ricci tensor, Ricci operator and scalar curvature respectively.

a b

,

are arbitrary constants such that

ab

≠

0

.

The quasi- conformal curvature tensor of a Sasakian manifold with respect to the quarter-symmetric metric connection

∇

~

is given by

(7.2)

{

2 (

1 )

}

{

(

,

)

(

,

)

},

)

1 (

~

]

~

)

,

(

~

)

,

(

)

,

(

~

)

,

(

~

[

)

,

(

~

)

,

(

~

Y

Z

X

g

X

Z

Y

g

b

n

a

n

r

Y

Q

Z

X

g

X

Q

Z

Y

g

Y

Z

X

S

X

Z

Y

S

b

Z

Y

X

R

a

Z

Y

X

W

−

+

−

+

−

+

(9)

& '( $)% $)*

where

R

~

,

S

~

,

Q

~

and

r

~

are the curvature tensor, Ricci tensor, Ricci operator and scalar curvature with respect to the quarter-symmetric metric connection respectively.

Let us consider an

n

-dimensional Sasakian manifold with respect to the quarter-symmetric metric connection

∇

,

which satisfies the condition

(7.3)

W

~

(

X

,

Y

)

Z

=

0

.

From (7.2) and (7.3), we have

(7.4)

{

2 (

1 )

}

{

(

,

)

(

,

)

}

0 .

)

1 (

~

]

~

)

,

(

~

)

,

(

)

,

(

~

)

,

(

~

[

)

,

(

~

=

−

+

−

+

−

+

Y

Z

X

g

X

Z

Y

g

b

n

a

n

r

Y

Q

Z

X

g

X

Q

Z

Y

g

Y

Z

X

S

X

Z

Y

S

b

Z

Y

X

R

a

Taking inner product on both sides of (7.4) by

ξ

we obtain

(7.5)

{

2 (

1 )

}

{

(

,

)

(

)

(

,

)

(

)}

0 .

)

1 (

)

1 (

2 )]

,

(

~

)

,

(

)

,

(

~

)

,

(

)

(

)

,

(

~

)

(

)

,

(

~

[

)

,

(

~

(

=

−

+

−

+

−

+

−

+

Y

Z

X

g

X

Z

Y

g

b

n

a

n

r

Y

S

Z

X

g

X

S

Z

Y

g

Y

Z

X

S

X

Z

Y

S

b

Z

Y

X

R

a

η

ξ

η

By virtue of (6.5), (2.14), (2.16) and (2.15) in (7.5), we get

(7.6)

.

0 )

(

)

,

(

2 )

(

)

,

(

2 )

(

)

,

(

)

(

)

,

(

)}

(

)

,

(

)

(

)

,

(

{

}

)

1 (

2 {

)

1 (

)

1 (

2 )

1

2 (

2 =

+

−

+

−

+

−

+

−

+

Y

X

Z

bd

X

Y

Z

bd

Y

Z

X

bS

X

Z

Y

bS

Y

Z

X

g

X

Z

Y

g

b

n

a

n

r

b

n

a

η

ϕ

η

ϕ

η

Putting

X

=

ξ

in (7.6) and using (2.10) and (2.4) we have

(7.7)

).

(

)

(

}

)

4

3 (

2 {

}

)

1 (

2 {

)

1 (

)

1 (

2 )

,

(

}

)

1 (

2 {

)

1 (

)

1 (

2 )

3

2 (

2 )

,

(

Z

Y

b

n

a

b

n

a

n

r

Z

Y

g

b

n

a

n

r

b

n

a

Z

Y

bS

η

−

+

−

+

−

+

−

+

−

+

−

+

=

−

Let

{

e

_i

:

i

=

1 ,

2 ,...,

n

}

be an orthonormal basis of the tangent space at any point of the manifold. Then putting

i

e

Z

Y

=

in (7.7) and taking summation for i ,

1 ≤

i

≤

n

, we get

(7.8)

,

(

2 )

0 )

2 (

)]

2

4

2 (

)

1 (

)[

1 (

2 ≠

−

+

−

+

−

+

−

=

if

a

n

b

n

a

n

b

n

a

n

r

.

In view of (7.7) and (7.8), we obtain

(7.9)

2

(2

3)

(2

9 6)

( , )

(

2)

(

2)

(

6 4)

( ) ( ),

(

2)

a

n

b

n

S Y Z

g Y Z

a

n

b

a n

b n

n

Y

Z

a

n

b

η

−

+

−

+

=

+

−

+

−

+

−

+

−

(10)

& '( $)% $)*

From equation (7.9), we get

(7.10)

S

(

Y

,

Z

)

=

α

g

(

Y

,

Z

)

+

βη

(

Y

)

η

(

Z

),

where

(7.11)

−

+

−

+

−

=

b

n

a

n

b

n

a

)

2 (

)]

6

9

2 (

)

3

2 (

2

α

and

(7.12)

−

+

−

+

−

=

b

n

a

n

b

n

a

)

2 (

)]

4

6

2 (

)

2 (

β

if

a

+

(

n

−

2 )

b

≠

0

.

Thus we arrive at the result:

Theorem 7.1: A quasi-conformally flat Sasakian manifold

(

M

n

,

g

)

with respect to the quarter-symmetric metric connection

∇

~

is an

η

-Einstein manifold provided that

a

+

(

n

−

2 )

b

≠

0

.

REFERENCES:

[1] Bhagawath, P., A pseudo-projective curvature tensor on a Riemannian manifold, Bull. of the Calcutta Math. Soci., 94(3) (2002), 163-166.

[2] Biswas, S. C. and De, U. C., Quarter-symmetric metric connection in an S. P. Sasakian manifold, common. Fac. Sci. Uni. Ank. Series, 46 (1997), 49-56.

[3] Blair, D. E., Contact manifolds in Riemannian Geometry, Lecture notes in Mathematics, Vol.509, Springer, Verlag, Berlin, 1976.

[4] De, U. C. and Shaikh , A. A., Complex manifolds and Contact manifolds, Narosa Publishing House Pvt.Ltd., 2009.

[5] Friedmann, A. and Schouten, J. A., Uber die Geometrie der halbsymmetrischen Ubertragung, Math. Zeitschr, 21(1924), 211-223.

[6] Golab, S., On semi-symmetric and quarter- symmetric linear connections, Tensor N. S., 29(1975), 249-254.

[7] Mishra, R. S. and Pandey, S. N., On quarter-symmetric metric F-connections, Tensor, N. S. 34(1980), 1-7.

[8] Mondal, A. K. and De, U. C., Some properties of a quarter-symmetric metric connection on a Sasakian manifold, Bull. of Math. Analysis and applications, 3(2009), 99-108.

[9] Mukhopadhyay, S., Roy, A. K. and Barua, B., Some properties of a quarter-symmetric metric connection on a Riemannian manifold, Soochow J. of Math.17(2) (1991), 205-211.

[10] Rastogi, S. C., On quarter- symmetric metric connection , C. R. Acad. Sci. Bulgar, 31(1978), 811-814.

[11] Szabo, Z. I., Structure theorems on Riemannian spaces satisfying

R X Y R

(

,

)

.

=

0,

I, The local version, J. Diff.

Geom., 17(1982), 531-582.

[12] Yano, K. and Imai, T., Quarter-symmetric metric connections and their curvature tensors, Tensor N.S., 38(1982), 13-18.

[13] Yano, K. and Kon, M., Structures on manifolds, Series in Pure Math., Vol.3, World Scientific, 1984.

[14] Yano, K. and Sawaki, S., Riemannian manifolds admitting a conformal transformation group, J. Diff. Geometry, 2(1968), 161-184.