Vol 9, No 4 (2018)

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Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 9(4), April-2018 66

ON TRANS-SASAKIAN MANIFOLDS SATISFYING CERTAIN CURVATURE CONDITIONS

RIDDHI JUNG SHAH*

Department of Mathematics, Janata Campus, Nepal Sanskrit University, Dang, Nepal.

(Received On: 27-02-18; Revised & Accepted On: 03-04-18)

ABSTRACT

T

he present paper deals with a study of trans-Sasakian manifolds satisfying certain curvature conditions on contact conformal curvature tensor. It is shown that the trans-Sasakian manifold satisfying the curvature conditions

,

0 ).

,

(

0

X

R

=

C

ξ

C

₀

(

ξ

,

X

).

C

₀

=

0

and

C

~

(

ξ

,

X

).

C

₀

=

0

is contact conformally semi-symmetric manifold.

Keywords: Trans-Sasakian manifold, contact conformal curvature tensor, semi-symmetric, concircular curvature tensor.

MSC 2010: 53C15, 53C25, 53D15.

1.INTRODUCTION

In [5], authors studied on the sixteen classes of almost Hermitian manifolds and their linear invariants. They considered unitary group

U

(

n

)

on a certain space

W

and studied that the representation of

U

(

n

)

on

W

has four irreducible components,

W

=

W

₁

⊕

W

₂

⊕

W

₃

⊕

W

₄

.

Among these components

W

₃

⊕

W

₄ corresponds to the class of Hermitian manifolds. A new class of almost contact metric structure, called trans-Sasakian structure was studied in [9] which is an analogue of a locally conformal Kaehler structure on an almost Hermitian manifold. An almost contact metric structure

(

ϕ

,

ξ

,

η

,

g

)

(where the symbols have their usual meanings) on

M

is trans-Sasakian [9] if

(

M

×

R

,

J

,

G

)

belongs to the class

W

₄

,

where

J

is the almost complex structure on

M

×

R

defined by

























₌

₋

dt

d

X

f

X

dt

d

f

X

J

,

ϕ

ξ

,

η

(

)

(1.1)

for any vector field

X

on

M

,

where

G

is the product metric on

M

×

R

.

_{Trans-Sasakian manifold is the} trans-Sasakian structure of type

(

α

,

β

),

where

α

and

β

are smooth functions on

M

.

Trans-Sasakian manifold of type

)

0 ,

(

),

0 ,

0 (

α

and

(

0 ,

β

)

are cosympletic [1],

α

-Sasakian and

β

-Kenmotsu manifold [2,7] respectively. Trans-Sasakian manifolds have been studied in [3], [10] and by others.

In [6], contact conformal curvature tensor field was introduced and defined by Jeong et al. in a

(

2 n

+

1 )

-dimensional Sasakian manifold.

2. PRELIMINARIES

Let

M

be a

(

2 n

+

1 )

-dimensional differentiable manifold with an almost contact metric structure

(

ϕ

,

ξ

,

η

,

g

),

where

ϕ

is a (1, 1) tensor field,

ξ

is a vector field,

η

is a 1-form and

g

is a compatible Riemannian metric such that [1]

,

)

(

)

(

2

η

ξ

ϕ

X

=

−

X

+

X

η

(

ξ

)

=

1 ,

ϕξ

=

0 ,

η

(

ϕ

X

)

=

0 ,

(2.1)

),

(

)

(

)

,

(

)

,

(

X

Y

g

X

Y

X

Y

g

ϕ

=

−

η

(2.2)

),

(

)

,

(

),

,

(

)

,

(

X

Y

g

X

Y

g

X

g

ϕ

=

−

ϕ

ξ

=

η

(2.3) for all

X

,

Y

∈

TM

.

The fundamental 2-form Φ of the almost contact metric structure

(

ϕ

,

ξ

,

η

,g

)

is defined as

), , ( ) , ( ) ,

(X Y =g X ϕY =−g ϕX Y

Φ (2.4)

Corresponding Author: Riddhi Jung Shah*

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Since ϕ is a skew-symmetric with respect to .g

An almost contact metric manifold M is called trans-Sasakian manifold if [9]

( )

∇X

ϕ

Y =

α

{g(X,Y)

ξ

−

η

(Y)X}+

β

{g(

ϕ

X,Y)

ξ

−

η

(Y)

ϕ

X}, (2.5) where ∇is Levi-Civita connection of Riemannian metric gand α,β are smooth functions on M.

From (2.5) it follows that

}, ) ( {

η

ξ

β

αϕ

ξ

X X X

X =− + −

∇ (2.6) ).

, ( ) , ( )

(∇_X

η

Y =−

α

g

ϕ

X Y +

β

g

ϕ

X

ϕ

Y (2.7)

In a (2n+1)-dimensional trans-Sasakian manifold M,the following relations hold [3]:

(

)

), ( ) ( ) ( ] ) ( ) ( [ 2

) ( ) ( ) ( ] ) ( ) ( [ )

, (

2 2 2

2

X Y X Y Y X X Y

Y X Y X Y X X Y Y

X R

ϕ

β

ϕ

α

ϕ

η

ϕ

η

αβ

ϕ

β

ϕ

α

η

β

α

ξ

+ +

− +

− −

=

(2.8)

), )( , ( ] ) ( )[ (

) )( , ( ) ( ] ) (

) , ( [ 2 ] ) ( ) , ( )[ (

) ,

( 2 2

β ϕ

ϕ ξ η β

α ϕ

ϕ α ϕ

η

ξ ϕ αβ η

ξ β

α ξ

grad Y X g X X Y

grad X Y g X Y X Y

X Y g X

Y Y

X g Y

X R

− −

+

+ +

−

+ −

− =

(2.9)

, 0 ) (

2αβ+ ξα = (2.10)

), )( 1 2 ( ) ) (( ) ( )] ( ) (

2 [ ) ,

(X

ξ

n

α

2

β

2

ξβ

η

X

ϕ

X

α

n X

β

S = − − − − − (2.11)

), , ) , ( ( ) ) , (

(R X Y Z g R X Y ξ Z

η =− (2.12) ,

0 ) , ( ( ) ) , ( ( ) ) , (

( ξ =η ξ ξ =η ξ ξ =

η R X Y R X R X (2.13)

). , ( )) ( (

) ) , (

(R

ξ

X Y

α

2

β

2

ξβ

g

ϕ

X

ϕ

Y

η

= − − (2.14)

In a (2n+1)-dimensional trans-Sasakian manifold if we put ϕ(gradα)=(2n−1)gradβ,then some of the above relations are reduced to

, 0 )

(ξβ = (2.15)

), ( ) (

2 ) ,

(X n 2 2 X

S

ξ

=

α

−

β

η

(2.16)

), , ( ) (

) ) , (

(R

ξ

X Y

α

2

β

2 g

ϕ

X

ϕ

Y

η

= − (2.17) },

) ( ){ (

) ,

( X 2 2 X X

R

ξ

=

α

−

β

η

ξ

− (2.18)

. ) , ( ) ,

(X ξ ξ Rξ X ξ

R =− (2.19)

We shall calculate all the results under the condition ϕ(gradα)=(2n−1)gradβ.

In a (2n+1)-dimensional trans-Sasakian manifold the contact conformal curvature tensor field C₀and the concircular

curvature tensor C~of type (1, 3) are defined by

0

1

( , ) ( , ) { ( , ) ( , ) ( , ) 2

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

( ) ( ) ( ) ( ) ( , )

( , ) ( , ) ( ) ( ,

C X Y Z R X Y Z S Y Z X S X Z Y g Y Z QX n

g X Z QY S X Z Y S Y Z X

X Z QY Y Z QX S X Z Y

S Y Z X g X Z Q Y g Y Z

η ξ η ξ

η η η η φ φ

φ φ φ φ φ

= + − +

− + −

+ − +

− + −

2

) ( )

2 ( , ) ( ) 2 ( , ) }

1 ( 2)

{2 2 }{ ( , )

2 22( 1) 2

1 (3 2)

( , ) 2 ( , ) } { 2 }

2 ( 1) 2

{

Q X

g X Y Q Z S X Y Z

n r

n n g Y Z X

n n n

n r

g X Z Y g X Y Z n

n n n

φ

φ φ φ φ

φ φ

φ φ φ φ

+ +

+

+ − − +

+

− − + + −

+

× 1 2 (3 2)

( , ) ( , ) } {4 5 2 }

2 ( 1) 2

{ ( ) ( ) ( ) ( ) ( , ) ( ) ( , ) ( ) } n r

g Y Z X g X Z Y n n

n n n

Y Z X X Z Y g Y Z X g X Z Y

η η η η η ξ η ξ

+

− − + + −

+

× − + −

(2.20)

(3)

} ) , ( ) , ( { ) 1 2 ( 2 ) , ( ) , ( ~

V Z U g U Z V g n n

r Z

V U R Z V U

C −

+ −

= (2.21)

for all X,Y,Z,U,V∈χ(M)respectively [4].

From (2.20), we have

}, ) ( ) ( ){ 2 (

) , ( ) ,

( 2 2

0 X Y R X Y Y X X Y

C ξ = ξ+ α −β − η −η _(2.22)

}, ) ( ) , ( ){ 2 (

) , ( ) ,

( 2 2

0 X Y R X Y g X Y Y X

C ξ = ξ + α −β − ξ −η _(2.23)

)}, ( ) , ( ) ( ) , ( ){ 2 (

) ) , ( ( ) ) , (

(C₀ X Y Z η R X Y Z α2 β2 g Y Z η X g X Z η Y

η = + − − − _(2.24)

, 0 ) ) , (

( ₀ ξ =

η C X Y (2.25) )}.

( ) ( ) , ( ){ 1 (

2 ) ) , (

( 2 2

0 X Y g X Y X Y

C ξ α β η η

η = − − − (2.26)

From (2.21), we also have

}. ) ( ) , ( { ) 1 2 ( 2 ) , ( ) , ( ~

V Z Z

V g n n

r Z

V R Z V

C ξ ξ ξ−η

+ −

= (2.27)

Definition: A (2n+1)-dimensional trans-Sasakian manifold M is said to be an Einstein manifold if its Ricci tensor S of type (0, 2) is of the form

) , ( ) ,

(X Y g X Y

S =λ (2.28) where λis a scalar function on M.

3. TRANS-SASAKIAN MANIFOLD SATISFYING

C

₀

(

ξ

,

X

).

S

=

0

Now, we prove the following result

Theorem 3.1: If a trans-Sasakian manifold Mof dimension (2n+1) satisfies the condition C0(ξ,X).S =0,then the

manifold is an Einstein manifold and its scalar curvature is r=2n(2n+1)(

α

2−

β

2).

Proof: Let Mbe a (2n+1)-dimensional trans-Sasakian manifold which satisfies the condition

. 0 ) , )( ). , (

(C0 ξ X S U V = (3.1) (3.1) implies that

. 0 ) ) , ( , ( ) , ) , (

(C₀ X U V +S U C₀ X V =

S ξ ξ (3.2)

Putting V =ξin (3.2) and using (2.16) and (2.23), we obtain

). , ( ) (

2 ) ,

(X U n 2 2 g X U

S =

α

−

β

(3.3)

Let {ei},i=1,2,...,2n+1be an orthonormal basis of the tangent space at any point of the manifold. Putting X =U =ei

in (3.3) and summing over i,1≤i≤2n+1,we get

). )(

1 2 (

2 +

α

2−

β

2 = n n

r (3.4)

In view of (3.3) and (3.4) it follows that the manifold Mis an Einstein manifold with scalar curvature

). )(

1 2 (

2 +

α

2−

β

2 = n n

r This completes the proof of the theorem.

C

~

(

ξ

,

X

).

C

₀

=

0

Theorem 4.1: Let M be a (2n+1)-dimensional trans-Sasakian manifold. If M satisfies the condition

, 0 ). , ( ~

0 = C X

C ξ then the manifold is contact conformally semi-symmetric. Proof: By definition we have

0

0 0 0

( , ) ( , ) ( ( , ) , ) ( , ( , ) ) ( , ) ( , ) 0 C ξ X C U V Z−C C ξ X U V Z−C U C ξ X V Z−C U V C ξ X Z=

(4)

Using (2.21) in (4.1) we get

0 0

0

( , ) ( , ) { ( , ( , ) ) 2 (2 1)

( ( , ) ) } ( , ) { ( , ) ( ) },

2 (2 1)

, ( , ) { ( , ) ( ) }

2 (2 1)

( , ) ( , ) { ( , ) ( ) } 0

2 (2 1) r

R X C U V Z g X C U V Z

n n

r

C U V Z X C R X U g X U U X V Z

n n

r

C U R X V g X V V X Z

n n

r

C U V R X Z g X Z Z X

n n

ξ ξ

η ξ ξ η

ξ ξ η

−

+

− − − −

+

− − −

+

− − − =

+





































Or,

0 0 0

0 0

0 0 0

0

[ ( , ) ( , ) ( ( , ) , ) ( , ( , ) )

( , ) ( , ) ] [ ( ( , ) ) 2 (2 1)

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

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

( ) ( , ) ] 0

R X C U V Z C R X U V Z C U R X V Z

r

C U V R X Z C U V Z X

n n

g X C U V Z g X U C V Z U C X V Z

g X V C U Z V C U X Z g X Z C U V

Z C U V X

ξ ξ ξ

ξ η

ξ ξ η

ξ η ξ

η

− −

− +

+

− + −

+ − +

− =

Or,

(

( , ). ₀

)

( , ) [ ( ₀( , ) ) 2 (2 1)

r

R X C U V Z C U V Z X

n n

ξ + η

+

0 0 0

0

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

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

( ) ( , ) ] 0.

g X C U V Z g X U C V Z U C X V Z

g X V C U Z V C U X Z g X Z C U V

Z C U V X

ξ ξ η

ξ η ξ

η

− + −

+ − +

− =

(4.2)

Putting Z=ξin (4.2) and taking inner product on both sides by ξwe get

. 0 ] ) , ( ( ) ) , ( , ( [ ) 1 2 ( 2 ) , ) , ( ). , (

( ₀ − ₀ − ₀ =

+

+ g X C U V C U V X

n n

r V

U C X R

g ξ ξ ξ ξ η (4.3)

Using (2.22) and (2.24) in (4.3), we get

)]. ) , ( ( ) ) , ( , ( [ ) 1 2 ( 2 ) , ) , ( ). , (

( ₀ g X R U V R U V X

n n

r V

U C X R

g ξ ξ ξ − ξ −η

+

+ (4.4)

In view of (2.8), (2.12) and (4.4) we obtain

. 0 ) , ( ). ,

(ξ X C0 U V ξ =

R (4.5) Equation (4.5) implies that the manifold is contact conformally semi-symmetric. This completes the proof of the theorem.

C

0

(

ξ

,

X

).

C

0

=

0

Theorem 5.1: Let Mbe a (2n+1)-dimensional trans-Sasakian manifold. The manifold M satisfying the condition

0 ). , ( 0 0 X C =

C ξ is contact conformally semi-symmetric if

. 0 ) , ( ) , ) , ( ( } ) , ( ) , ( ) , ( 2 ){

(

α

2−

β

2 g V Z X −g X V Z−g X Z V −g R

ξ

V Z X

ξ

−R X V Z =

Proof: Let Mbe a trans-Sasakian manifold of dimension (2n+1)which satisfies the condition

, 0 ) , ( ). , ( ₀

0 X C U V Z=

C ξ then by definition we have

0( , ) 0( , ) 0( 0( , ) , ) 0( , 0( , ) ) 0( , ) 0( , ) 0 .

C ξ X C U V Z−C C ξ X U V Z−C U C ξ X V Z−C U V C ξ X Z = (5.1)

Using (2.23) in (5.1) we get

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

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

) ) , ( , ( )[ 2 (

) , ( ). , (

0 0

0

0 0

0

0 2

2 0

− +

−

+ −

−

− − +

V U C Z X g Z X U C V Z U C V X g

Z V X C U Z V C U X g X Z V U C

Z V U C X g Z

V U C X R

ξ η

ξ

η ξ

η

ξ β

α ξ

(5)

. 0 )] ) , ( ( ) ( ) ) , ( ( ) (

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

) ) , ( , ( )[ 2 (

) , ) , ( ). , ( (

0 0

0 2

2 0

= −

+

− +

− −

− − +

X V U C Z Z X U C V

Z U C V X g Z V X C U

Z V C U X g Z V U C X

Z V U C X g Z

V U C X R g

η η η

η

ξ η

η η

ξ η η

η

β α ξ ξ

(5.3)

Putting U =ξin (5.3) and using (2.23), (2.24) and (2.26), we get

. 0 )}] , ( ) (

) , ( ) ( 2 ) , ( ) ( ){ (

) ) , ( (

) , ) , ( ( )[ 2 (

) , ) , ( ). , ( (

2 2

2 2 0

= +

− −

+ +

− − +

Z X g V

Z V g X V

X g Z Z

V X R

X Z V R g Z

V C X R g

η

η η

β α η

ξ β

α ξ ξ ξ

(5.4)

Equation (5.4) implies that

]. ) , (

) , ) , ( ( } ) , ( ) , (

) , ( 2 ){ )[(

2 (

) , ( ). ,

( 2 2 2 2

0

Z V X R

X Z V R g V Z X g Z V X g

X Z V g Z

V C X R

−

− −

−

− −

− =

ξ ξ

β α β

α ξ

ξ

(5.5)

From (5.5) it follows that the manifold M is contact conformally semi-symmetric if the right hand side vanishes i.e., if

2 2

(α −β ){2 ( ,g V Z X) −g X V Z( , ) −g X Z V( , ) }−g R( ( , ) ,ξ V Z X)ξ−R X V Z( , ) =0 . (5.6)

This completes the proof of the theorem.

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[2] Blair, D. E., Riemannian geometry of contact and sympletic manifolds, Birkhauser, Boston, 2002.

[3] De, U. C. and Tripathi, M. M., Ricci-tensor in 3-dimensional trans-Sasakian manifolds, Kyungpook Math. J. 43(2)(2003), 247-255.

[4] De, U. C. and Shaikh, A. A., Differential geometry of manifolds, Narosa Publishing House, New Delhi, India, 2007.

[5] Gray, A. and Hervella, L. M., The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Math. Pura Appl. 123(4) (1980), 35-58.

[6] Jeong, J. C., Lee, J. D., Oh, G. H. and Pak, J. S., On the contact conformal curvature tensor, Bull. Korean Math. Soc. 27(2)(1990), 133-142.

[7] Kenmotsu, K., A class of almost contact Riemannian manifolds, Tohoku Math. J. 24(1972), 93-103.

[8] Kitahara, H., Matsuo, K. and Pak, J. S., A conformal curvature tensor on Hermitian manifolds, Bull. Korean Math. Soc. 27(1990), 27-30.

[9] Obina, J. A., New classes of almost contact metric structures, Pub. Math. Debrecen 32(1985), 187-193. [10] Shah, R. J., On trans-Sasakian manifolds, Kathmandu Univ. J. of Sci., Eng. and Tech. 8(1) (2012), 81-87.

Source of support: Nil, Conflict of interest: None Declared.