J. Raj. Academy of Physical Sciences, Vol.10. N.1. (2011,} 63-72(A Quarterly International Journal)

ON -RECURRENT TRANS-SASAKIAN MANIFOLDS

By

S.Yadav and D.L.Suthar

Department of Mathematics

Alwar Institute of Engg. &Technology,M.I.A.Alwar-301030, India

Email:prof_sky16@yahoo.com,dd_suthar@yahoo.co.in

Abstract

The objective of the present paper is to study -recurrent trans –Sasakian manifolds

1. Introduction

In 1977, T.Takahashi [1] introduction the notion of locally -symmetric Sasakian manifolds and

studied their several properties. Many authors like De and Pathak[2] Venktasha and Bagewadi[4]

and Shaikh and De[3] have extended this notion to three-dimensional Kenmotsu,trans-Sasakian

and an LP-Sasakian manifolds respectively. In 2003, U.C.De, A.A.Shaikh and S.Biswas[5],

introduced the notion of –recurent Sasakian manifolds which generalized the notion of locally

–symmetric Sasakian manifolds and studied their several properties. In 2009, G.T.Sreenivasa,

Venkatesha,C.S.Gagewadi and K.Naganagoud [6] studied this notion in Lorentzian, –

Kenmotsu manifolds and obtain several results. Motivated by the above studies in this paper we

define –recurrent trans-Sasakian manifold, -Ricci-symmetric and pseudo projective –

recurrent trans-Sasakian manifold and tree-dimensional case of -Ricci-symmetric also

consider and obtain some result.

2. Preliminaries

An )12( n -dimensional smooth manifold 12 nM is said to be an almost contact metric manifold

[7] if it admits a )1,1( –tensor field , a vector , a-form and a Riemannian metric g ,which

satisfy

(2.1) (a) 0 (b) 0)( X (c) )(2 XXX

(2.2) (a) 1)( (b) )(),( XXg (c) ),(),( YXgYXg

(d) )()(),(),( YXYXgYXg

for all vector fields YX , on 12 nM An almost contact metric manifold 12 nM ),,,( g is said to be

trans-Sasakian manifold[10] if ),,( GJM belong to the class4W of Hermitian manifolds

whrere J ,is a almost complex structure on M defined by

dt

dZfZ

dt

dfZJ )(,,

for any vector field Z on 12 nM and smooth function f on M and G is the product metric on

M .This may be states by the condition[8]

(2.3) XYYXgXYYXgYX )(),()(),())((

where , are smooth functions on 12 nM and such a structure is said to be the trans-Sasakian

structure of type ),( from (2.3) it follows that

(2.4) )(XXXX

(2.5) ),(),()( YXgYXgX

In a trans-Sasakian manifold 12 nM ),,,( g the following relations hold [9]

(2.6)

)()()()()(2

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

2

222

XYXYYXXY

YXYXYXXYYXR

(2.7)

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

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

)(),()(),()()),((),),(( 22

ZXZXgYZYgXZYZYg

YZXgYZYgXZXgY

YZXgXZYgZYXRZYXRg

(2.8) XXYR )()(),( 22

(2.9) )()()()(),()(),( 222 XXXYYXgYXR

(2.10) ))(12())(()()()(2),( 22 XnXXnXS

(2.11) )()(2),( 22 nS

(2.12) 02)( n

(2.13) ))(12()()()(2 22 gradngradnQ

(2.14)

))(()))((12(

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

2

2222

XXn

YXnYXgnYXS

where R is the curvature tensor of type )3,1( of the manifold and Q is the symmetric

endomorphism of the tangent space at each point of the manifold corresponding to the Ricci

tensor S ,that is ),(),( YXSYQXg for any vector fields YX , on 12 nM

3. -recurrent trans-Sasakian manifolds

Definition3.1.A trans-Sasakian manifold is said to be locally –symmetric if

(3.1) 0),)((2 ZYXRW

for all vector fields WZYX ,,, orthogonal to [1].

Definition3.2.A trans-Sasakian manifold is said to be –recurrent manifold if there exist a non-

zero 1–form A such that [11]

(3.2) ZYXRWAZYXRW ),()(),)((2

for arbitrary vector fields WZYX ,,,

If the 1–form A vanishes, and then the manifold reduces to a –symmetric manifold

Definition3.3.A trans-Sasakian manifold is said to be locally -Ricci symmetric if the Ricci

operator Q satisfies [11]

(3.3) 0))((2 XQW

for all vector fields XW, on 12 nM .

If XW, are orthogonal to , then the manifold is said to be locally -Ricci symmetric.

Let us consider -recurrent trans-Sasakian manifold 12 nM . Then by virtue of (2.1-c) and (3.1),

we get

(3.4) ZYXRWAZYXRZYXR WW ),()()),)(((),)((

From which it follows that

(3.5) ),),(()()()),)(((),),)(( UZYXRgWAUZYXRUZYXRg WW

Let 12....3,2,1, niei be the orthonormal basis of the tangent space at any point of the

manifold. Then putting ieUX in (3.4) and taking summation over i , ,121 ni we get

(3.6) ),()()(),)((),)((12

1

ZYSWAeZYeRZYS i

n

i

iWW

The second term of (3.6) for Z takes the form

),(),,),)(( iiiW egeYeRg

also

),),((),),((),),((),),((),),)(( WWiWiWiW YeRgYeRgYeRgYeRgYeRg

at 12 nMp

Since ie is an orthonarmal basis 0 iX e at 12 nMp using (2.4)(2.6)(2.8)(2.9),we get

))(()())(),()()((),),(( 22 YYegYYeRg WWiWWi

Thus we obtain

(3.7)

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

))(),()()(),),((),),)(( 22

WW

WiWiWiW

YeRgY

YegYYeRgYeRg

In view of 0),),((),),(( ii eYRgYeRg

Then 0),),((),),(( WiiW YeRgYeRg

This implies (3.7)

(3.8) ),),((),),((),),)(( WWiiW YeRgYeRgYeRg

Using (2.4) and applying the skew-symmetry of R ,we get

(3.9)

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

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

YWRgeYWRgeYRgW

eYWRgeYWRgYeRg

ii

iiiW

Using (3.9) in (3.6),we get

(3.10) ),),((),),((),),(()(2

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

YWRgeYWRgeYRgW

eYWRgeYWRgYSWAYS

ii

iiW

Since ),(),(),)((),)(( WWWW YSYSZYSYS

Using (2.4) and (2.10) in above, we get

(3.11)

),(),()())(()(())(()()()12(

)()())(()()(2),)(( 22

WYSWYSWYYXWYn

YWYnYS

WW

WW

From (3.10) and (3.11), we get

(3.12)

)()(()()(())((2))(()())(1(2

)()()1(),(),()()(2),(),( 22

WAYWYYXWYn

WYWYgWYgnWYSWYS

WW

We can state the following result.

Theorem 3.1.In a –recurrent trans-Sasakian manifold 1),,( 12 ngM n then the equation of Ricci

tensor is given by (3.12)

Now from (3.4), we have

(3.12) ZYXRWAZYXRZYXR WW ),()()),)(((),)((

From (3.12) and the Bainchi identity, we get

(3.13) 0)),(()()),(()()),(()( ZXWRYAZYXRXAZYXRWA

By virtue of (2.7) and (3.13) ,we get

(3.14)

0)}()(),()(

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

)}()(),(){(

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

)()(),(){(

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

22

22

22

ZWZWgW

XZZXgWXZWgWZYgYA

ZYZYgW

WZZWgYWZYgYZWgXA

ZXZXgY

ZYZYgXYZXgXZYgWA

putting ieZY in(3.14) and taking summation over i , ,121 ni we get

(3.15) )()()()()()( 2222 WXAXWA , for all vector fields XW,

Replacing X by in (3.15), we get

(3.16) )()()( WWA , for all vector fields .W

where )(),()( gA , being the vector fields associated to the 1–form A that is

),()( XgXA

We can state the following result.

Theorem 3.2. In a –recurrent trans-Sasakin manifold 1),,( 12 ngM n , the characteristic vector

field and the vector field associated to the 1–form A are co-directional and the the 1–form A

given by )()()( WWA .

4. Locally -Ricci symmetric trans-Sasakian manifold

Let us consider that the manifold 12 nM is –Ricci symmetric. Then from (2.1-c) (3.3), we have

(4.1) 0)))((())(( YQYQ XX

Taking the inner product of (4.1) with Z ,we have

(4.2) 0)()))((()),)((( ZYQZYQg XX

On simplifying, we get

(4.3) 0)()))((()),((),( ZYQZYQgZYS XXX

Replacing Y in (4.3) and using (2.4)(2.10), we get

(4.4) 0)()))(((),()()(2),()(),(),( 22 ZYQZgnZSXZXSZXS XX

Replacing Z by Z in (4.4), we have

(4.5) 0),()()(2),()(),(),( 22 ZgnZSXZXSZXS X

Using(2.10)(2.14),we get

(4.6) XnXZXgZXgZXgnZXS )(12())(),(),(),()()(2),( 22

We can state the following result.

Theorem 4.1. If a trans-Sasakian manifold 1),,( 12 ngM n is locally -Ricci-symmetric then the

equation of Ricci tensor is given by (4.6)

5. Pseudo-projective -recurrent trans-Sasakian manifold

Definition5.1.A trans-sasakian manifold is said to be pseudo-projective -recurrent manifold if

there exist a non-zero 1-form Asuch that

(5.1) ZYXPWAZYXPW ),(~

)(),)(~

(2

for arbitrary vector fields .,,, WZYX

where P~

is a pseudo-projective curvature tensor defined as

(5.2) YZXgXZYgbn

a

n

rYZXSXZYSbZYXaRZYXP ),(),(

2)12(),(),(),(),(

~

where ba , are constant such that 0, ba , R is the curvature tensor, S is the Ricci tensor and r is

the scalar curvature

If the 1–form A vanishes, then the manifold reduces to a locally pseudo-projective -symmetric

manifolds.

We consider that the manifold 12 nM to be a pseudo-projective –recurrent then (2.1-c) and

(5.1), we have

(5.3) ZYXPWAZYXPZYXP WW ),(~

)()),)(~

((),)(~

(

From which it follows that

(5.4) ),),(~

()()()),)(~

((),),)(~

( UZYXPgWAUZYXPUZYXPg WW

Let 12....3,2,1, niei be the orthonormal basis of the tangent space at any point of the

manifold. Then putting ieUX in (5.4) and taking summation over i , ,121 ni we get

(5.5) 0)(),)((),)((),)()(2( YZSZYSbZYSanb WWW

Replacing Z by and using (2.10) ,we get

(5.6)

0)())(12())(()())}(12()){((2

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

2

22

YWnbWbYXnXb

YWnnbYSW

Using (3.11) in (5.6), we get

(5.7)

)()])(12())(()})(12(){((2[

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

))())(()(2}[()12({

)()())((2)((2))()(2)()12((),(

2

22

2222

YWnbWbXnXb

WYSWYYY

Ynbna

YWnnbnbnaWYS

WW

W

We can state the following results.

Theorem5.1. If a trans-Sasakian manifold 1),,( 12 ngM n is a pseudo-projective –recurrent then

the following relation (5.7) hold.

6. Three-dimensional locally -Ricci symmetric trans-Sasakian manifold

In three-dimensional Riemannian manifold, we have

(6.1) YZXgXZYgr

YZXSXZYSQYZXgQXZYgZYXR ),(),(2

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

Where Q is the Ricci operator i.e. ),(),( YXSYQXg and r is the scalar curvature

Putting ZY in (6.1) and using (2.2-b) (2.6) (2.10) for 3n , we get

(6.2)

XXXn

XXgradngrad

Xr

n

Xnnr

QX

)(2))(12(

))(()())(12()(

)(2

)()()(4

)())(12()(2)(2(2

2222

2222

Using (2.12) in (6.2), we get

(6.3)

))(12())(()())(12()(

)(2

)()()(4

)())(12()(2)(2(2

2222

2222

XnXXgradngrad

Xr

n

Xnnr

QX

Taking the vector field YX , orthogonal to

(6.4)

))(12())(()())(12()(2)(2(2

2222 XnXXnnr

QX

Taking covariant derivative (6.4) with respect to W ,we get

(6.5) XWdrXQW )(2

1)(

Operating both sides by2 , we get

(6.6) XWdrXQW22 )(

2

1)(

We can state the following result.

Theore.6.1.A three-dimensional trans-Sasakian manifold 1),,( 12 ngM n is locally-Ricci

symmetric if and only if the scalar curvature r is constant.

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