Vol 3, No 12 (2012)

Available online throug

ISSN 2229 – 5046

CURVATURE TENSORS EQUIPPED

WITH AN INTEGRATED CONTACT METRIC STRUCTURE MANIFOLD

SHALINI SINGH*

Department of Applied Mathematics, JSS Academy of Technical Education, Noida-201301, India

(Received on: 31-07-12; Revised & Accepted on: 22-08-12)

ABSTRACT

I

n the present paper, I have defined an integrated contact metric structure manifold [6] admitting semi-symmetric metric connexion [4] in

M

_n∗ and the form of curvature tensor

R

of the manifold relative to this conexion has been derived. Several useful theorems and results have also been derived in this manifold.

Key words:

C

∞-manifold, integrated contact structure, integrated contact metric structure, Riemannian connexion, Semi-symmetric metric connexion.

AMS Mathematics Subject Classification No: 53.

1. Introduction

Let n

M

be a differentiable manifold of differentiability class

C

∞. Let there exist in n

M

a vector valued

C

∞ linear

function Ф, a

C

∞- vector field

η

and a

C

∞-one form

ξ

such that (1.1)

Φ

2

( )

X

=

a X

2

−

c

ξ

( )

X

η

(1.2)

( )

η

=

0

(1.3)

G X Y

(

,

)

=

a G X Y

2

(

,

)

−

c

ξ

( ) ( )

X

ξ

Y

Where

Φ

( )

X

=

X

,

a

is a non-zero complex number and

c

is an integer.

Let us agree to say that

Φ

gives to n

M

a differentiable structure define by algebraic equation (1.1). We shall call

(

Φ

, , , ,

η

a c

ξ

)

as an integrated contact structure.

Remark 1.1: The manifold n

M

equipped with anintegrated contact structure

(

Φ

, , , ,

η

a c

ξ

)

will be called an integrated contact structure manifold.

Remark 1.2: The

C

∞-manifold n

M

satisfying (1.1), (1.2) and (1.3) is called an integrated contact metric structure

manifold

(

Φ

, , , , ,

η

a c G

ξ

)

Agreement 1.1: All the equations which follows will hold for arbitrary vector fields

X Y Z

, , ...

etc.

It is easy to calculate in

M

_n that

(1.4)

( )

a

2

c

ξ η =

(1.5)

Φ

( )

X

=

0

and

(1.6)

G X

(

,

η

)

def

ξ

( )

X

Corresponding author:SHALINI SINGH*

Remark 1.3: The integrated contact metric structure manifold

(

Φ

, , , , ,

η

a c G

ξ

)

gives an almost norden contact

metric manifold [5], Lorentzian para-contact manifold [2] or an almost Para contact Riemannian manifold [1]

according as

(

a

2

= −

1,

c

=

1 ,

) (

a

2

=

1,

c

= −

1

)

or

(

a

2

=

1,

c

=

1

)

Agreement 1.2: An integrated contact metric structure manifold will be denoted by

M

_n. In the sequel, arbitrary vector

fields will be denoted by

X Y Z

, , ,...

etc.

If we define

(1.7) '

Φ

(

X Y

,

)

=

G X Y

(

,

) (

=

G X Y

,

)

Where '

Φ

is a tensor field of the type (0, 2) then it is easy to see that (1.8) '

Φ

(

X Y

,

)

= Φ

'

(

Y X

,

)

Which shows that '

Φ

is symmetric in

X

and

Y

. Also we have (1.9) '

Φ

(

X Y

,

)

=

a

2 '

Φ

(

X Y

,

)

Definition 1.1: A

C

∞-manifold n

M

satisfying (1.10)

D

_X

η = Φ

X

will be denoted by

M

_n∗ where

D

is the Riemannian connexion in n

M

corresponding to the Riemannian metric

G

. It

is easy to calculate in

M

_n∗, we have

(1.11)

(

D

_X

ξ

)( )

Y

= Φ

`

(

X Y

,

)

2. Semi-symmetric metric connexion in

C

∞-manifold

n

M

∗:

Definition 2.1: Let

D

be a Riemannian connexion in

M

_n∗. We consider a semi-symmetric metric connexion

B

in

n

M

∗

(2.1)

B Y def D Y

_{X X}

+

ξ

( )

Y X

−

G X Y

(

,

)

η

The above equation is equivalent to

(2.2)

B Y

_X

=

D Y

_X

+

H X Y

(

,

)

where

(2.3)

H X Y

(

,

)

d e

f

ξ

( )

Y X

−

G X Y

(

,

)

η

Let

S

be the torsion tensor of the semi-symmetric metric connexion

B

then (2.1) and (2.2) imply that

(2.4)a

S X Y

(

,

)

=

ξ

( )

Y X

−

ξ

( )

X Y

(2.4)b

S X Y

(

,

)

=

H X Y

(

,

)

−

H Y X

(

,

)

We define

(2.5)a '

H X Y Z

(

, ,

)

d e

f G H X Y

(

(

,

)

,

Z

)

(2.5)b '

S X Y Z

(

, ,

)

d e

f G S X Y

(

(

,

)

,

Z

)

Operating

G

on both the sides of (2.3) and using (2.5)a, we get

(2.6) '

H X Y Z

(

, ,

)

=

ξ

( ) (

Y G X Z

,

) ( ) (

−

ξ

Z G X Y

,

)

Operating

G

on both the sides of (2.4)a and using (2.5)b, we get

/IJMA- 3(12), Dec.-2012.

Operating

G

on both the sides of (2.4)b and using (2.5)a and (2.5)b, we get

(2.8) '

S X Y Z

(

, ,

)

=

`

H X Y Z

(

, ,

)

−

`

H Y X Z

(

,

,

)

3. Curvature Tensor in n

M

∗:

Let

K

be the curvature tensor corresponding to the Riemannian connexion

D

in

M

_n∗ and

R

be the curvature tensor

corresponding to the semi-symmetric metric connexion

B

in

M

_n∗. From (2.1), we have

(3.1)

B Z

_Y

=

D Z

_Y

+

ξ

( )

Z Y

−

G Y Z

(

,

)

η

Taking the covariant derivative of the above equation with respect to the connexion

B

along the vector field

X

, we get

B B Z

X Y

=

B

X

(

D Z

Y

) ( )

+

ξ

Z B Y

X

+

{

(

B

X

ξ

)( ) (

Z

+

ξ

B Z

X

)

}

Y

−

G B Y Z

(

_X

,

)

η

−

G Y B Z

(

,

_X

) (

η

−

B G Y Z

_X

)(

,

)

η

−

G Y Z

(

,

)(

B

_X

η

)

Using (2.1) and then (1.6) in the above equation, we get

(3.2)a

B B Z

_{X Y}

=

D D Z

_{X Y}

+

ξ

(

D Z X

_Y

)

−

G X D Z

(

,

_Y

)

η ξ

+

( )

Z D Y

_X

+

ξ

( ) ( )

Z

ξ

Y X

−

ξ

( ) (

Z G X Y

,

) (

η

+

D

_X

ξ

)( )

Z Y

+

ξ

(

D Z Y

_X

)

−

ξ

( ) (

Y G X Z

,

)

η

+

ξ

( ) ( )

Z

ξ

X Y

−

G X Z

(

,

) ( )

ξ η

Y

−

G D Y Z

(

_X

,

)

η

−

G X Y

(

,

) ( )

ξ

Z

η

−

G Y D Z

(

,

_X

)

η

−

G Y X

(

,

) ( )

η ξ

Z

−

G X Z

(

,

) ( )

ξ

Y

η

−

(

B G Y Z

_X

)(

,

)

η

−

G Y Z X

(

,

)

−

G Y Z

(

,

) ( )

ξ η

X

+

G Y Z

(

,

) ( )

ξ

X

η

Interchanging

X

and

Y

in the above equation, we get

(3.2)

B B Z

_{Y X}

=

D D Z

_{Y X}

+

ξ

(

D Z Y

_X

)

−

G Y D Z

(

,

_X

)

η ξ

+

( )

Z D X

_Y

+

ξ

( ) ( )

Z

ξ

X Y

−

ξ

( ) (

Z G Y X

,

)

η

+

(

D

Y

ξ

)( )

Z X

+

ξ

(

D Z X

Y

)

−

ξ

( ) (

X G Y Z

,

)

η

+

ξ

( ) ( )

Z

ξ

Y X

−

G Y Z

(

,

) ( )

ξ η

X

−

G D X Z

(

Y

,

)

η

−

G Y X

(

,

) ( )

ξ

Z

η

−

G X D Z

(

,

Y

)

η

−

G X Y

(

,

) ( )

η ξ

Z

−

G Y Z

(

,

) ( )

ξ

X

η

−

(

B G

_Y

)(

X Z

,

)

η

−

G X Z Y

(

,

)

−

G X Z

(

,

) ( )

ξ η

Y

+

G X Z

(

,

) ( )

ξ

Y

η

Replacing

Y

by

[

X Y

,

]

in (3.1) and using

[

X Y

,

]

=

D Y

_X

−

D X

_Y , we get

(3.2)c

B

_{[X Y, ]}

Z

=

D

_{[X Y, ]}

Z

+

ξ

( )(

Z

D Y

_X

) ( )(

−

ξ

Z

D X

_Y

)

−

G D Y Z

(

_X

,

)

η

−

G D X Z

(

_Y

,

)

η

Subtracting (3.2)b, (3.2)c from (3.2)a and using

K X Y Z d

(

, ,

)

f D D Z

e

_{X Y}

−

D D Z

_{Y X}

−

D

_{[X Y, ]}

Z

, we get

(

, ,

)

(

, ,

) (

_X

)( )

3

( ) (

,

)

R X Y Z

=

K X Y Z

+

D

ξ

Z Y

−

ξ

Y G X Z

η

−

(

D

_Y

ξ

)( )

Z X

−

G Y Z X

(

,

)

+

(

B G

_Y

)(

X Z

,

)

η

+

G X Z Y

(

,

)

where

(

, ,

)

X Y Y X _[X Y, _]

R X Y Z

d

f B B Z

e

−

B B Z

−

B

Z

Agreement 3.1: We consider that the fundamental 2-form

_`

Φ

is closed in

M

_n∗ i.e.

(3.4)

(

D

X

`

Φ

)(

Y Z

,

) (

+

D

Y

`

Φ

)(

Z X

,

) (

+

D

Z

`

Φ

)(

X Y

,

)

=

0

Theorem 3.1: In

M

_n∗, we have

(3.5)

(3.6)

(

D

X

`

Φ

)(

Y

,

η

)

= −

G X Y

(

,

)

Proof: We know that

`

Φ

(

Y Z

,

) (

=

D

Y

ξ

)( )

Z

Taking the covariant derivative on both the sides of the above equation with respect to the vector field

X

, we have

(3.7)a

(

D

_X

`

Φ

)(

Y Z

,

) (

=

D D

_{X Y}

ξ

)( )

Z

− Φ

`

(

D Y Z

_X

,

)

Interchanging

X

and

Y

in the above equation, we get

(3.7)b

(

D

_Y

`

Φ

)(

X Z

,

) (

=

D D

_{Y X}

ξ

)( )

Z

− Φ

`

(

D X Z

_Y

,

)

Subtracting (3.7)b from (3.7)a and using

D Y

_X

−

D X

_Y

=

[

X Y

,

]

, we have

(

D

X

`

Φ

)(

Y Z

,

) (

−

D

Y

`

Φ

)(

X Z

,

) (

=

D D

X Y

ξ

)( )

Z

−

(

D D

Y X

ξ

)( )

Z

− Φ

`

(

[

X Y

,

]

,

Z

)

Using (1.11) in the above equation, we get

(

D

X

`

Φ

)(

Y Z

,

) (

−

D

Y

`

Φ

)(

X Z

,

) (

=

D D

X Y

ξ

)( ) (

Z

−

D D

Y X

ξ

)( )

Z

−

(

D

[X Y, ]

ξ

)

( )

Z

=

G Z K X Y

(

,

(

, ,

η

)

)

=

`

K X Y

(

, , ,

η

Z

)

= −

`

K X Y Z

(

, , ,

η

)

= −

G K X Y Z

(

(

, ,

)

,

η

)

= −

ξ

(

K X Y Z

(

, ,

)

)

Using (3.4) in the above equation, we get (3.5).

We have

`

Φ

(

Y

,

η

)

=

0

(

D

_X

`

)(

Y

,

η

)

`

(

Y D

,

_X

η

)

⇒

Φ

= − Φ

Using (1.10) in the above equation, we get

(

D

X

`

Φ

)(

Y

,

η

)

= − Φ

`

(

Y X

,

)

Using (1.7) in the above equation, we get (3.6).

Theorem 3.2: In

M

_n∗, we have

(3.8)

(

D

Z

`

Φ

)(

X Y

,

)

=

c





ξ

( ) (

X G Y Z

,

) ( ) (

−

ξ

Y G X Z

,

)





Proof: From (1.9), we have

(

)

2

(

)

,

,

`

Φ

X Y

=

a

`

Φ

X Y

Taking the covariant derivative on both the sides of the above equation with respect to the vector field

Z

, we have

(

)

(

)

(

(

)

)

(

(

)

)

2

(

)(

)

,

,

,

,

`

`

`

`

Z Z Z Z

D

Φ

X Y

+ Φ

D

Φ

X Y

+ Φ

X

D

Φ

Y

=

a

D

Φ

X Y

Using (1.7) in the above equation, we get

(

)

(

)

(

(

)( )

)

(

(

)( )

)

2

(

)(

)

,

,

,

,

`

`

Z Z Z Z

/IJMA- 3(12), Dec.-2012.

Using (1.1) in the above equation, we get

(

)

(

)

2

(

)(

)

( )

(

)

( )

(

)

,

,

,

,

0

`

`

Z Z

D

Φ

X Y

+

a

D

Φ

X Y

+

c

ξ

Y G Z X

−

c

ξ

X G Z Y

=

Using (1.3) in the above equation, we get

(

)

(

)

2

(

)(

)

2

( ) (

)

2

( ) (

)

,

,

,

,

0

`

`

Z Z

D

Φ

X Y

+

a

D

Φ

X Y

+

a c

ξ

Y G X Z

−

a c

ξ

X G Y Z

=

We define

Ρ

(

X Y Z

, ,

)

d e

f a

2

(

D

Z

`

Φ

)

(

X Y

,

)

(

)

2

(

)(

)

2

( ) (

)

2

( ) (

)

, ,

Z

`

,

,

,

Q X Y Z

d ef a

D

Φ

X Y

+

a c

ξ

Y G X Z

−

a c

ξ

X G Y Z

Then

Ρ

(

X Y Z

, ,

)

+

Q X Y Z

(

, ,

)

=

0

Also in consequence of (3.6), we have

(

X Y Z

, ,

)

Q X Y Z

(

, ,

)

,

Q X Y Z

(

, ,

)

(

X Y Z

, ,

)

Ρ

=

= Ρ

All these equations are satisfied by

Ρ = =

Q

0

i.e

a

2

(

D

_Z

`

Φ

)(

X Y

,

)

= −

a c

2

ξ

( ) (

Y G X Z

,

)

+

a c

2

ξ

( ) (

X G Y Z

,

)

i.e.

(

D

Z

`

Φ

)(

X Y

,

)

=

c





ξ

( ) (

X G Y Z

,

) ( ) (

−

ξ

Y G X Z

,

)





This proves the theorem.

Theorem 3.3: In

M

_n∗, we have

(3.9)

`

K X Y Z U

(

, , ,

)

−

`

K X Y Z U

(

, , ,

)

+

`

K X Y Z U

(

, , ,

)

+

`

K X Y Z U

(

, , ,

)

+

(

D K

_η

`

)

(

X Y Z U

, , ,

)

=

0

Proof: From (3.5), we have

(3.10)a

(

D

Y

`

Φ

)(

Z U

,

)

=

`

K Z U Y

(

, , ,

η

)

Taking the covariant derivative on both the sides of the above equation with respect to the vector field

X

and using (1.10) and (3.5), we have

(

D D

_{X Y}

`

Φ

)(

Z U

,

) (

=

D K

_X

`

)(

Z U Y

, , ,

η

)

+

`

K Z U D Y

(

, ,

_X

,

η

)

+

`

K Z U Y X

(

, , ,

)

+

`

K D Z U Y

(

_X

, , ,

η

) (

+

Z D U Y

,

_X

, ,

η

)

−

(

D Z U Y

_X

, , ,

η

) (

−

Z D U Y

,

_X

, ,

η

)

i.e.

(3.10)b

(

D D

X Y

`

Φ

)(

Z U

,

) (

=

D K

X

`

)(

Z U Y

, , ,

η

)

+

`

K Z U D Y

(

, ,

X

,

η

)

+

`

K Z U Y X

(

, , ,

)

Interchanging

X

and

Y

in the above equation, we get

(3.10)c

(

D D

_{Y X}

`

Φ

)(

Z U

,

) (

=

D K

_Y

`

)(

Z U X

, ,

,

η

)

+

`

K Z U D X

(

, ,

_Y

,

η

)

+

`

K Z U X Y

(

, ,

,

)

Replacing

Y

by

[

X Y

,

]

in (3.10)a, we have

(3.10)d

(

D

_[X Y_{, ]}

`

Φ

)

(

Z U

,

)

=

`

K Z U X Y

(

, ,

[

,

]

,

η

)

Subtracting (3.10)c and (3.10)d from (3.10)b, we have

i.e.

`

K X Y Z U

(

, , ,

)

−

`

K X Y Z U

(

, , ,

)

=

(

D K

X

`

)(

Z U Y

, , ,

η

)

+

`

K Z U D Y

(

, ,

X

,

η

)

(

, , ,

)

(

)(

, ,

,

)

`

K Z U Y X

D K

Y

`

Z U X

η

+

−

−

`

K Z U D X

(

, ,

_Y

,

η

)

−

`

K Z U X Y

(

, ,

,

)

−

`

K Z U

(

, ,

[

X Y

,

]

,

η

)

`

K X Y Z U

(

, , ,

)

−

`

K X Y Z U

(

, , ,

)

=

(

D K

X

`

)(

Z U Y

, , ,

η

)

+

`

K Z U Y X

(

, , ,

)

−

(

D K

Y

`

)(

Z U X

, ,

,

η

)

−

`

K Z U X Y

(

, ,

,

)

i.e.

`

K X Y Z U

(

, , ,

)

−

`

K X Y Z U

(

, , ,

)

−

`

K Z U Y X

(

, , ,

)

+

`

K Z U X Y

(

, ,

,

)

(

D K

_X

`

)(

Z U Y

, , ,

η

) (

D K

_Y

`

)(

Z U X

, ,

,

η

)

=

−

(

D K

X

`

)(

Y

, , ,

η

Z U

) (

D K

Y

`

)(

η

,

X Z U

, ,

)

=

−

Using Bianchi’s second identity [3] in the above equation, we get (3.9).

Corollary 3.1: In n

M

∗, we have

(3.11)a

a K X Y Z U

2

`

(

, , ,

)

+

`

K X Y Z U

(

, , ,

)

= −

c

ξ

( ) (

U K X Y Z

`

, , ,

η

)

(

, , ,

)

`

K X Y Z U

−

−

`

K X Y Z U

(

, , ,

)

−

(

D K

η

`

)

(

X Y Z U

, , ,

)

(3.11)b

`

K X Y Z U

(

, , ,

)

−

`

K X Y Z U

(

, , ,

)

+

`

K X Y Z U

(

, , ,

)

+

a K X Y Z U

2

`

(

, , ,

)

−

c

ξ

( ) (

X K

`

η

, , ,

Y Z U

)

+

(

D K

_η

`

)(

X Y Z U

, , ,

)

=

0

(3.11)c

`

K X Y Z U

(

, , ,

)

−

`

K X Y Z U

(

, , ,

)

+

a K X Y Z U

2

`

(

, , ,

)

−

c

ξ

( ) (

Y K X

`

, , ,

η

Z U

)

+

`

K X Y Z U

(

, , ,

)

+

(

D K

_η

`

)

(

X Y Z U

, , ,

)

=

0

Proof: Barring

U

on both the sides of (3.9) and using (1.1), we get (3.11)a.

Barring

X

on both the sides of (3.9) and using (1.1), we get (3.11)b.

Barring

Y

on both the sides of (3.9) and using (1.1), we get (3.11)c.

Theorem 3.4: In n

M

∗, we have

(3.12)a

Ric Y Z

(

,

)

+

a Ric Y Z

2

(

,

)

=

c

ξ

( )

Y Ric

(

η

,

Z

)

−

(

D Ric Y Z

_η

)(

,

)

(3.12)b

Ric Y Z

(

,

)

+

a Ric Y Z

2

(

,

)

=

c

ξ

( )

Z Ric

(

η

,

Y

)

−

(

D Ric Y Z

_η

)( )

,

Proof: Equation (3.9) can be written as

(

)

(

, ,

,

)

(

(

, ,

)

,

)

(

(

, ,

)

,

)

G K X Y Z U

−

G K X Y Z U

+

G K X Y Z U

+

G K X Y Z U

(

(

, ,

)

,

)

+

G

(

(

D K

_η

)

(

X Y Z U

, ,

)

,

)

=

0

Which is equivalent to

(

, ,

)

(

, ,

)

(

, ,

)

K X Y Z

−

K X Y Z

+

K X Y Z

+

K X Y Z

(

, ,

)

+

(

D K

_η

)

(

X Y Z

, ,

)

=

0

Contracting the above equation with respect to the vector field

X

, we get

(3.13)

Ric Y Z

( )

,

+

Ric Y Z

(

,

)

+

(

D Ric Y Z

_η

)

(

,

)

=

0

Barring

Y

in the above equation and using (1.1), we get (3.12)a.

/IJMA- 3(12), Dec.-2012.

Corollary 3.2: The scalar curvature of n

M

∗-manifold is constant along the vector field

η

.

Proof: Equation (3.13) can be written as

(

,

) (

,

)

(

(

)

( )

,

)

0

G RY Z

+

G RY Z

+

G

D R Y

_η

Z

=

Where

R

is the Ricci tensor of the type

( )

1, 1

.

Using (1.7) in the above equation, we get

(

,

)

(

,

)

(

(

)( )

,

)

0

G RY Z

+

G RY Z

+

G

D R Y

_η

Z

=

which is equivalent to

RY

+

RY

+

(

D R Y

η

)

( )

=

0

Contracting the above equation with respect to the vector field

X

, we get

( )

D r

_η

= ⇒

0

η

r

=

0

This proves the theorem.

Theorem 3.5: In n

M

∗, we have

(3.14)a

`

K X Y Z U

(

, , ,

)

−

`

K X Y Z U

(

, , ,

)

=

c G U Y



(

,

) (

`

Φ

X Z

,

)

−

G Y Z

(

,

) (

`

Φ

X U

,

)

−

G U X

(

,

) (

`

Φ

Y Z

,

)

+

G X Z

(

,

) (

`

Φ

Y U

,

)



(3.14)b

`

K X Y Z U

(

, , ,

)

−

a

2

`

K X Y Z U

(

, , ,

)

=

c



`

Φ

(

Y U

,

) (

`

Φ

X Z

,

)

− Φ

`

(

Y Z

,

) (

`

Φ

X U

,

)



(

) (

)

(

) (

)

2

,

,

,

,

a

G Y Z G X U

G X Z G Y U



−

_

−

_

Proof: We have

(3.15)a

`

K X Y Z U

(

, , ,

)

−

`

K X Y Z U

(

, , ,

)

+

`

K X Y Z U

(

, , ,

)

+

`

K X Y Z U

(

, , ,

)

=

(

D K

_X

`

)(

Y

, , ,

η

Z U

) (

−

D K

_Y

`

)(

X

, , ,

η

Z U

)

From (3.5) and (3.8), we have

(3.15)b

`

K X Y Z

(

, , ,

η

)

=

c

_



ξ

( ) (

X G Y Z

,

) ( ) (

−

ξ

Y G Z X

,

)



_

The above equation can be written as

`

K Y

(

, , ,

η

Z U

)

=

c





ξ

( ) (

Z G U Y

,

) ( ) (

−

ξ

U G Y Z

,

)





Taking the covariant derivative of the above equation with respect to the vector field

_X

, we have

(

D K

X

`

)(

Y

, , ,

η

Z U

)

+

`

K Y X Z U

(

,

, ,

)

=

c





(

D

X

ξ

)( ) (

Z G U Y

,

) (

−

D

X

ξ

)( ) (

U G Y Z

,

)





Using (1.11) in the above equation, we get

(3.16)a

(

D K

_X

`

)(

Y

, , ,

η

Z U

)

=

`

K X Y Z U

(

, , ,

)

+

c G U Y

_



(

,

) (

`

Φ

X Z

,

)

−

G Y Z

(

,

) (

`

Φ

X U

,

)



_

Interchanging

X

and

Y

in the above equation, we get

(3.16)b

(

D K

Y

`

)(

X

, , ,

η

Z U

)

= −

`

K X Y Z U

(

, , ,

)

+

c G U X





(

,

) (

`

Φ

Y Z

,

)

−

G X Z

(

,

) (

`

Φ

Y U

,

)





Subtracting (3.16)b from (3.16)a, we have

(3.17)

(

D K

X

`

)(

Y

, , ,

η

Z U

) (

−

D K

Y

`

)(

X

, , ,

η

Z U

)

=

`

K X Y Z U

(

, , ,

) (

+

K X Y Z U

, , ,

)

+

c G U Y



(

,

) (

`

Φ

X Z

,

)

−

G Y Z

(

,

) (

`

Φ

X U

,

)

(

,

) (

`

,

)

(

,

) (

`

,

)

G U X

Y Z

G X Z

Y U

From (3.15)a and (3.17), we get (3.14)a.

Barring

U

on both the sides of (3.14)a and using (1.1) and (1.7), we get

(

)

(

)

( ) (

)

2

, , ,

, , ,

, , ,

`

K X Y Z U

−

a K X Y Z U

`

+

c

ξ

U K X Y Z

`

η

=

c



`

Φ

(

Y U

,

) (

`

Φ

X Z

,

)

− Φ

`

(

Y Z

,

) (

`

Φ

X U

,

)

(

) (

)

(

) (

)

2 2

,

,

,

,

a G X U G Y Z

a G X Z G Y U



−

+

_

Using (3.15)b in the above equation, we get (3.14)b.

REFERENCES

[1] Adati T. and Matsumoto K., On almost paracontact Riemannian manifold, T.R.U. Maths., 13(2) (1977), 22-39.

[2] Matsumoto K., On Lorentzian Paracontact manifolds, Bull. Yamogata Univ. Nat. Sci., 12(1989), 151-156.

[3] Mishra R. S., Structure on a differentiable manifold and their application, Chandrma Prakashan, Allahabad, India. (1984).

[4] Sharfuddin A. and Hussain S. I., Semi-symmetric metric connections in almost contact manifolds. Tensor, N. S., 30, (1976), 133-139.

[5] Singh S. D. and Singh D., Tensor of the type (0, 4) in an almost Norden contact metric manifold, Acta Cincia Indica, India, 18 M (1) (1997), 11-16.

[6] Singh Shalini, Certain affine connexions on an integrated contact metric structure manifold, Adv. Theor. Appl. Math., 4 (1-2) (2009), 11-19.