Vol 2, No 12 (2011)

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GENERALIZED BI-RECURRENT WEYL SPACES

1

_{Girish Dobhal* and}

_{Asha Ram Gairola}

1,2

Department of Computer Application, Graphic Era University, Dehradun, INDIA

E-mail: [email protected], [email protected]

(Received on: 21-10-11; Accepted on: 07-11-11)

ABSTRACT

In this paper, we have studied the generalized bi-recurrent Weyl spaces. The bi-recurrent properties of Weyl Conformal curvature, Weyl projective curvature and Weyl concircular curvature tensor have been studied and some relation between them has been derived.

Mathematics Subject Classification: 53A40

Keywords: Generalized Weyl space, Bi-recurrent Weyl space.

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1. Introduction:

An-n-dimensional differentiable manifold is said to be Weyl space if it has a symmetric connection and a symmetric conformal metric tensor preserved by . Accordingly, in local coordinates there exists a covariant vector field (complementary vector field) satisfying the condition [1], [2], [3],

(1.1)

The above equation can be extended to

(1.2)

where are the connection coefficients of the symmetric connection and are defined as

(1.3)

Moreover, under the renormalization condition

(1.4)

of the metric tensor , the covariant vector field is transformed by the law

(1.5) where is a scalar function defined on . We denote such a Weyl space by or .

An n-dimensional differential manifold having an anti-symmetric connection and anti-symmetric metric tensor preserved by is called generalized Weyl space [4]. It is denoted by ! .

For such a space, in local co-ordinate system, the compatibility condition are

(1.6)

Where are the components of a covariant vector field, called the complementary vector field of the ! space. Using the concept of covariant differentiation ([5], [6]), the compatibility condition of (1.6) can be written as

1

_{Girish Dobhal*,}

Page: 2503-2507

% & ! & ! " '

" " (1.7)

where " are the connection coefficient of the anti-symmetric connection and are obtain from the compatibility condition as

" #$% % % & (1.8)

Now putting

% #$% % % & (1.9)

We obtain

" % (1.10)

Where and % are respectively the coefficient of a Weyl connection and the torsion tenor of ! space and are expressed as

#'" " ( " (1.11) and

% ₎ #'" ₎ "₎ ( "_{$ )&} (1.12)

square bracket stands for anti-symmetry.

The components of mixed curvature tensor and Ricci tensor of ! are respectively

" " " "₎ ") "₎") (1.13)

" "*_* (1.14)

on the other hand scalar curvature of ! is defined by

" " (1.15)

It is easy to see that curvature tensor " of ! can be written as

" + % ₎ (1.16)

Where the tensors + and % ₎ are defined respectively

+ ₎ ) ₎ ) (1.17)

% % % %₎%) %₎ %) %₎%) (1.18)

The curvature tensor of ! satisfies the relation [6].

" ₎ "₎ (1.19)

" ₎ " ₎ "₎ ' %₎ % ₎ %₎ % % % ₎ % %₎ ( (1.20)

" ₎ "_{) )}" '"_,)%, "_, %₎, "_, %,₎( (1.21)

The Weyl Conformal curvature- , Weyl projective curvature and Weyl concircular curvature tensor . in ! are respectively given by [7][8],

- / '0 /$ & 0 /$ & /$ & /$ & 0 / (

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% & ! & ! "

/ _1## 3/ 0 / 0 4 (1.23)

. / 2_1# 3 0 0 4 (1.24)

From (1.22) and (1.24) we have

- . ₁ '0 ._{$ &} 0 ._{$ &} ._{$ &} ._{$ &} 0 . (

#₁ '0 . 0 . . . ( (1.25)

where . / 2 (1.26)

2. Bi-Recurrent Weyl Spaces:

Definition: 2.1 If the curvature tensor " of ! satisfies the condition

" _{*5 *5}" (2.1)

where _*5 is a non-zero covariant tensor field, then ! is called bi-recurrent. Such a space is denoted by ! .

Definition: 2.2 If the curvature tensor - of ! satisfies the condition

- _{*5 *5}- (2.2)

where _*5 is a non-zero covariant tensor field, then ! is called generalized Weyl space with bi-recurrent Weyl Conformal curvature tensor. We denote such a space by - ! .

Definition: 2.3 If the curvature tensor . of ! satisfies the condition

. _{*5 *5}. (2.3)

where _*5 is a non-zero covariant tensor field, then ! is called generalized Weyl space with bi-recurrent Weyl Concircular curvature tensor. We denote such a space by . ! .

Definition: 2.4 If the curvature tensor of ! satisfies the condition

_{*5 *5} (2.4)

where _*5 is a non-zero covariant tensor field, then ! is called generalized Weyl space with bi-recurrent Weyl Projective curvature tensor. We denote such a space by ! .

Theorem: 2.1 The necessary and sufficient condition for a Generalized Weyl space ! to be ! is that it should be . ! .

Proof: Let ! satisfies the relation (2.3) then (2.3) in view of (1.24) reduces to

/ _{*5 *5}/ 32 67186724

1# 3 0 0 4 (2.5)

contracting the indices and we have

/ _{*5 *5}/ 32 67186724 (2.6)

Now differentiating (1.23) covariantly with respect to 9 and then again differentiating the result thus obtained with respect to : we have

_*5 / _*5 #

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% & ! & ! " (

Above equation in view of (2.5) and (2.6) reduces to

_{*5 *5}/ 32 67186724

1# 3 0 0 4

#

1# ; *5/ 0

32 67186724 _<

0 ; _*5/ 32 67186724 < (2.8)

_*5;/ #

1# 3/ 0 / 0 4< (2.9)

_*5

therefore the space is ! .

Conversely let the space be ! then (2.4) in view of (1.23) gives

/ *5 _1## 3/ *50 / *50 4 *5$/ _1## 3/ 0 / 0 4&

(2.10) transecting (2.10) by we obtain

/ _{*5 *5}/ #3 _*5/ / _*54 (2.11)

Therefore we have

0 / _*5 0 / _{*5 *5}30 / 0 / 4 2 _*530 0 4

2 6730 0 4 (2.12) Substituting (2.12) in (2.10) we have

/ _*5 2 67_1# 30 0 4 _*5;/ 2_1# 30 0 4<

which reduces to

. _{*5 *5}.

Theorem: 2.2 The necessary and sufficient condition for a Generalized Weyl space ! to be - ! is that it should be . ! .

Proof: Let ! satisfies the relation (2.3) then (2.3) in view of (1.24) reduces to (2.5) and (2.6).

Now differentiating (1.24), (1.25) and (1.26) covariantly with respect to 9 and then again differentiating the result thus obtained with respect to : we have respectively

. _*5 / _*5 2 67

1# 3 0 0 4 , (2.13)

- _*5 . _{*5 1} $0 ._{$ & *5} 0 ._{$ & *5} ._{$ & *5} ._{$ & *5}

0 ._{$ & *5}& #₁ '0 . _*5 0 . _*5 . _*5 . _*5( (2.14)

and . _*5 / _*5 2 67 (2.15)

Equation (1.14) in view of (2.13) and (2.15) reduces to

- _*5 / _*5 2 67_1# 3 0 0 4 ₁ $0 =/ _*5 2 67 > 0 =/ _*5 2 67 >

=/ _*5 2 67 > =/ _*5 2 67 > 0 =/ _*5 2 67 >

#

1 $0 =/ *5

2 67 _{> 0 =/}

*5 2 67 > =/ *5 2 67 >

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% & ! & ! " )

Equation (2.16) in view of (2.5), (1.24) and (2.6) reduces to

- _{*5 *5}@.

1 '0 .$ & 0 .$ & .$ & .$ & 0 . (

#

1 '0 . 0 . . . (A

which reduces to

- _{*5 *5}

-Conversely, let the space be - ! then (2.2) in view of (1.25) reduces to

. _{*5 1} '0 ._{$ & *5} 0 ._{$ & *5} ._{$ & *5} ._{$ & *5} 0 . _*5(

#₁ '0 . _*5 0 . _*5 . _*5 . _*5( _*5@. ₁ $0 ._{$ &} 0 ._{$ &}

.$ & .$ & 0 . & #₁ '0 . 0 . . . (A

which reduces to

. *5 *5. $0 .$ & *5 *5.$ & 0 .$ & *5 *5.$ & *5

._{$ & *5 *5}.$ & .$ & *5 *5.$ &

0 . _{*5 *5}. & #₁ $0 . _{*5 *5}. 0 . _{*5 *5}.

.$ & *5. . *5 *5. & (2.17)

Equation (2.17) in view of (1.26)(2.6) and (2.15) reduces to

. *5 *5.

therefore the space is . ! .

REFERENCES:

[1] Zlatanov, G. Nets in the n-dimensional space of Weyl, C.R. acad. Bulgare. Sci. 41, (1988), 29-32.

[2] Tsareva, B. and Zlatanov, G. On the geometry of the nets in the n-dimensional space of Weyl, J. Geom. 18, (1990), 182-197

[3] Norden,, A. Affinely connected spaces, GRMFL Moscow, (1976).

[4] Murgescu,, V. Espaces de Weyl generalized, Rev. Rour. Math. Pure. et. Appl., XX, (2),(1970),293

[5] Murgescu, V. Espaces de Weyl generalized, Bul. Inst. Pol. de Jassy., 1970.

[6] Zeren Akgun, L. On generalized weyl Spaces, Bul. Cal. Math. Soc., 91, (4), (1999), 267-278.

[7] Miron, R. Mouvements confermes elans les espaces et B , Tensor N.S. 19,(1968),33-41.

[8] Singh U.P. and Singh A. K., On Kaehlerian conharmonic recurrent and Kaehlerian conharmonic symmetric space, Acc. Naz. Dei Lincei, Rend, 62(2), (1977). 173-179.

[9] K.S. Rawat, and Girish Dobhal, On the bi-recurrent Bochner curvature tensor,Journal of the Tensor Society,

1(2007), 33-40.

[10] K.S. Rawat, and Girish Dobhal, Bi-recurrent and bi-symmetric Concircularcurvature tensor of Riemannian space, Jour. PAS, 15(2009), 54-60.