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RECURRENT FINSLER SPACES THEOREM IN PROJECTIVE RECURRENT SPACES OF THIRD ORDER

Shailesh Kumar Srivastava and A.K. Mishra

Sri Ram Swaroop Memorial College of Engineering & Management, Faizabad Road, Tiwariganj, Lucknow

E-mail: Shaileshks82@gmail.com

Abstract: The present paper deals with study of recurrence of generalized second order in a Finsler space

F

n* equipped with non-symmetric connection. We have tried to make study of special birecurrent and special generalized birecurrent

F

n* of first and second kinds and have obtained results of significance. An attempt has also been made to study the recurrence of third order in such a Finsler space with special reference to the first curvature tensor

( , )

i

R

jk

x x 

of *

F

n .

Keywords: Finsler spaces, special birecurrent, Ricci-recurrent spaces.

INTRODUCTION

The recurrent Finsler spaces have been studied by Mishra and Pande [3], Sen, R.N.

[1], Chaki and Chaudhary [2], have introduced the Ricci-recurrent spaces of second order and studied the properties of the recurrence vector and tensor field and the curvature tensor field

i

H

jkh in an n-dimensional Finsler space Fn equipped with Berwald’s symmetric connection coefficient

G

ijk

( , ) x x

. Ray, A. K. [4] has defined the generalized 2-recurrent Riemannian space. An attempt to extend the theory of generalised 2-recurrent curvature tensor to Finsler geometry has been made by Pande and Khan [5] and many more. Pande studied recurrent Finsler spaces of third order having symmetric connection parameter

G

ijk and Berwald’s covariant derivative defined in it. Pandey and Mishra, R.B. [6] has carried out a comparative study of various types of recurrent Finsler spaces by using Berwald’s and Cartan’s curvature tensors.

Received Jan 18, 2016 * Published Feb 2, 2016 * www.ijset.net

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1. SPECIAL

R

+- BIRECURRENT

F

n*:

Differentiating

R

ijkh

λ R

ijkh

+

+

=



 ⊕ - covariantly with respect to

x

m and thereafter using

h h

ijk m ijk m

R λ R

+ +

+ +

=



 , we get

(1.1)

0

λ

 + m

=

or

0

h

R

ijk +

=

. Therefore, we can state:

THEOREM (1.1):

If a

R

+-recurrent

F

n* of first order be special

R

+-birecurrent of second kind then such an

*

F

n is ether flat one or in such an

F

n* the recurrence vector is a covariant constant.

Let us now suppose that the special birecurrent

F

n* of first kind be also 1-recurrent then in such a case we can easily state the following:

COROLLARY (1.1):

If a special birecurrent

F

n* of first kind be also 1-recurrent then such an

F

n* is either flat one or in such an

F

n* the recurrence vector is a ⊕ - covariant constant.

Commutating ijkh m ijkh

R

m

λ R

+ +

+ +



=

 with respect to the indices



and m and thereafter using the relevant commutation formula, we get

(1.2)

h r r h h h h r

p

ijk pjk ijk r m rjk i m irk j m

r

R R x R R R R R R

+ + + + + + + +

 

∂ + − − −

 

 

  

 

0

h r h

p

ijr k m ijk p m

R R R N

+ + +



− |



=

.

While deriving (1.2), we have taken into account that the space under consideration is also 1- recurrent. Exactly a similar result can be obtained if we start our discussion from

h h

ijk m ijk m

R λ R

+ +

+ +

=



 and proceed in a like manner. Therefore, we can state:

(3)

THEOREM (1.2):

In a special birecurrent

F

n* of first and second kinds, (1.2) always holds. Transvecting

h h

ijk m m ijk

R λ R

+ +

+ +



=

 and ijkh ijkh

m m

R λ R

+ +

+ +

=



 by the metric tensor

g

ip, we get

(1.3) a) jpkh jpkh

m m

R

+ + +

= λ



R

+ +



And b)

R

+ jpkh+ m +

= λ

m

R

+ jpkh +

 

respectively. Conversely the transvection of (1.3a) and (1.3b) by the associate tensor

g

ip of

the metric tensor

g

ij yield ijkh m ijkh

R

m

λ R

+ +

+ +



=

 and ijkh ijkh

m m

R λ R

+ +

+ +

=



 respectively.

Then, the above conditions are equivalent to the conditions (1.3a) and (1.3b) respectively.

Therefore, we can state:

THEOREM (1.3):

A special

R

+-birecurrent space of first kind and special

R

+-birecurrent space of second kind may be characterized by the equation (1.3a) and (1.3b) respectively.

Contracting ijkh m ijkh

R

m

λ R

+ +

+ +



=

 and ijkh ijkh

m m

R λ R

+ +

+ +

=



 with respect to the indices i and h, we get

(1.4) a) jk jk

m m

R

+ + +

= λ



R

+ +



And b)

R

+ jk + m +

= λ

m

R

+ jk +

 

respectively. Thus, the Ricci tensor

R

jk of a special

R

+-birecurrent

F

n* of first and second kinds satisfy (1.4a) and (1.4b) respectively. Conversely if the Ricci tensor of a Finsler space

*

F

n satisfies (1.4a) or (1.4b) then the space need not be special

R

+-birecurrent of first and second kinds respectively, Therefore, we can state:

THEOREM (1.4):

The Ricci tensor

R

jk of a special

R

+-birecurrent

F

n* of first and second kinds respectively satisfy (1.4a) and (1.4b) and conversely if the Ricci tensor of a Finsler space

F

n* satisfies

(4)

(1.4a) or (1.4b) then the space need not be special

R

+-birecurrent of first and second kinds respectively. Transvecting (1.4a) and (1.4b) by

g

jp, we get

(1.5) a)

p p

k k

m m

R

+ + +

= λ



R

+ +

 ,

And b)

p p

k m m k

R

+ + +

= λ R

+ +

 .

Contracting the indices p and k in (1.5a) and (1.5b) and thereafter using the fact that

p

R

p+

= R

+ , we get (1.6) a)

m m

R λ R

+ +

+ + +

=





and b) m

R

+ + m +

= λ R

+ +

 . Therefore, we can state:

THEOREM (1.5):

In a special birecurrent

F

n* of first and second kinds the contracted tensor R also behaves accordingly.

2. GENERALISED 2-RECURRENT

R F

+ n*:

DEFINITION (2.1):

A Finsler space

F

n* is said to be generalized birecurrent of first kind if the curvature tensor

h

R

ijk +

satisfies the relation

(2.1)

h h h

ijk mn n ijk m mn ijk

R β R α R

+ + +

+ +

= +

where

β

n and

α

mn are respectively the non zero associated vector and associated tensor of recurrence.

We commute (2.1) with respect to the indices m and n and thereafter use the relevant commutation formula and get

(2.2)

h r r h h r h r h

r

irijk mn ijk rmn rjk imn irk jmn nm ijk r

R R R R R R R R N R

+ + + + + + + +

+

− + − − +   −

 

( )

h r h h h

ijr kmn n ijk m m ijk n mn nm ijk

R R

+ +

β R

+ +

β R

+ +

a a R

+

− = − + −

.

(5)

With the help of (2.2) we can easily verify that

a

mn is non-symmetric however if we assume that

a

mn is symmetric recurrence tensor and

R

ijkh is a first order recurrent curvature tensor with respect to given associated vector of recurrence then from (2.2), we get

(2.3)

h r r h h n h r h n

rijk mn ijk rmn rjk imn irk jmn ijr kmn

R R R R R R R R R R

+ + + + + + + + + +

− + − − − +

0.

h r r

R

ijk

N

nm

β

+

+ =

Contracting (2.3) with respect to the indices h and k and using the relevant commutation formula

X

i +

|

jk

X

i +

|

kj

= − ∂ ( 

m

X

i

) R

pjkm

x 

p

+ X R

m mjki

+ X

i +

|

m

N

kjm , we get

(2.4)

0

r r r

ij mn rj imn ir jmn ij r

r

R

+

R

+

R R

+ +

R R

+ +

β

r

R N

+ nm

 

− ∂   − − + =

 



.

The following identities can be easily obtained (2.5) a)

R

k

R x

jk j

+ +

= 

, b)

R x x

+ij

 

i j

= R x

+ j



j

= ( n 1 ) R

+

c) r

R

j

x

j

( n 1 )

r

R R

r

+ + +

 

∂ = − ∂ −

 

 

  

.

Transvecting (2.4) successively by

x

i and

x

j and thereafter using ( 2.5a, b, c), we get (2.6)

r mn r

r

R R

+ +

β

r

R N

+ nm

 

∂ =

 

 



. Therefore, we can state:

THEOREM (2.1):

In a

R

+ generalized 2-

F

n* (2.6) always holds. Transvecting (2.1) by

x

i, we get

(2.7)

h h h

jk mn n jk m mn jk

R

+ +

= β R

+ +

+ a R

+ .

Commutating (2.7) with respect to the indices m and n and using the relevant commutation formula, we get

(2.8)

− ∂ ( 

r

R

hjk

) R

+rmn

+ R R

+rjk +hrmn

R R

+hrk +rjmn

R R

+hjr +hkmn

+

( )

h h h h

jk r nmr n jk m m jk n mn nm jk

R

+ +

N β R

+ +

β R

+ +

α α R

+

 

+   = − + −

 

.

(6)

Differentiating (2.8) ⊕ - covariantly with respect to

x

 and then transvecting the result thus obtained, we get

(2.9)

( ) ( )

h h h

k k k

mn nm

R

mn nm

R

n

R

m

α − α

+ +

+ α − α

+ +

+ β

+ +

+

  

h h h

k k k

n

R

m m

R

n m

R

n

β

+ + +

β

+ +

β

+ + +

+ − −

  

h r h r h h

r j

jk mn jk mn rmn k

r

R x R

r

R x R R R

+ + + + + +

+ + +

   

= − ∂   − ∂   + +

 



 

 

   

r h h r h r h r

k rmn rk mn rk mn r kmn

R R R R R R R R

+ + + + + + + +

+ + + +

+ − − − −

   

h r h h

r r

r kmn k r nm k r nm

R R R N R N

+ + + +

+ + + +

− + +

  .

Contracting (2.9) with respect to the indices h and k and then using (2.1) and the set of equations given by (2.5), we get

(2.10)

R

+ r

( β



N

nmr

+ N

nmr +

) + α

r

R N

+ nmr



( n 1 )( α

mn

α

nm

)

+

( β α

n m

β α

m n

) R

+

=   − − + − +

  

( α

mn

α

nm

) R

+

( β β

n

β

n +

) R

+ m

+ −



+



+ −



( β β

m

β

m +

) R

+ n

r

R R

+

+rmn

+

− + +   ∂   

  

  

  



.

Allowing a cyclic interchange of the indices

 , m

and n in (2.10) and thereafter adding all the three equations thus obtained, get

(2.11) [ [ ] [

]

r r

r nm r nm

R

+ r

β R

+

α N R N

+ +

 

+ +

 

 



( n 1 ) { γ

[mn+ ]

β γ

[n m ]

} R

+

γ

[mn

R

+ ]

β

[n+

R

+m]

= −  |



+



+ |



+ |



[ ] [

r

m

R

n r

R R

mn

β

+ +

+

+

+

− +   ∂   

  

  

 

| | 

,

(7)

where, (2.12)

. dif

mn mn nm

γ = α − α

. Therefore, we can state:

THEOREM (2.2):

In a Finsler space

F

n* with generalized birecurrent curvature tensor

h

R

ijk +

, the identity (2.11) always holds. Differentiating (2.7) partially with respect to

x

i we get

(2.13)

i

R

+hjk + mn

= ∂

i

R

+hjk m

β

n

+ R

+hjk + m

(

i

β

n

) +

   

  

(

i

α

mn

) R

+hjk

α

mn

i

R

+hjk

+ ∂ +  ∂ 

 

 

.

Using (a)

x 

i +

|

k

= = 0 x 

i

|

k, (b) i i h

jk hjk

R = R x 

, (c) i i h

j hj

R = R x 

,

(d)

R

hjki

= − R

hkji

, R

ijk

= − R

kji , (e)

N

ijk

= − N

kji

= Γ − Γ

ijk ikj , (f)

Γ

ihjk

= ∂ Γ 

h ijk

. And (2.1) the identity (2.13) assumes the following form

(2.14)

h r r h

q h h r

iqjk mn jk n irm jk m irn rk n ijm

x R R R R

+ + + +

+

+ + +

+  Γ + Γ − Γ −



h h h h

r r r

rk m ijn jr n ikm jr m ikn rjk n

R R R R

+ + + +

+ + +

+

− Γ − Γ − Γ −  +

r h h

q r p q r p

rqjk n ipm rjk m rqjk m ipn

x R x R x R x

+ + +

+

 

+ +

  

+  Γ −  +  Γ  +

    

   

r h h h

h r r

jk irm n rk ijm n jr ikm n rjk

R R R R

+ + + +

+ + +

+ Γ − Γ − Γ  +

h h s

q r p r h

rqjk ipm n jk n imn jk rsm

x R x R R

+ + +

+ +

 

+  Γ − Γ −  Γ −

 

 

h h h h

s s q s t r p

sk rjm js rkm sjk sqjk rtm ipn

R R R x R x x

+ +

+ +

 

− Γ − Γ −  +  Γ  Γ

  

  

h r h h

q h r r

iqjk jk rk jr

n

x R

m

R

irm

R

ijm

R

ikm

β

+ + + + +

=  + Γ − Γ − Γ −



(8)

( )

h h h

q r p

rjk rqjk ipm jk m n

R

+

x R

+

x R

+ +

β

  

−  +  Γ  + ∂ +

   



  

( α

mn

) R

+hjk

α

mn

x R

q +iqjkh

+ ∂



+

 

.

Commutating (2.14) with respect to the indices m and n and then writing hijk n ijkh

m

β

Γ

+

= Γ

, we get

(2.15)

[ ] [ ] [ ]

h h h

q t s r p

iqjk mn sqjh rt m ip n iqjh m n

x R

+ +

R

+

x Γ Γ

< >

xR

+ +

β −

  

- [ ]

h iqjh mn

R

+

α

[ [ ]

1 2

r s t s r p

s s

j r i mn rj m rt m ip n

R N R R x x

+ +

+

< >

 

= ∂ +  Γ + Γ  Γ +

 

  

[ ] [ ]

.

j i mn j m i n

R

+

α R

+ + < >

β

+ ∂  + ∂ 

Therefore, we can state:

THEOREM (2.3):

In a Finsler space

F

n* with generalized birecurrent curvature tensor

R

ijkh the identity given by (2.15) always holds provided

Γ

hijk be supposed to be first order recurrent with respect to the associated vector of recurrence.

DEFINITION (2.2):

A Finsler space

F

n* is said to be generalized birecurrent of second kind if the curvature tensor

R

ijkh satisfies the following relation

(2.16)

h h h

ijk mn m ijk n mn ijk

R

+ +

= β R

+ +

+ α R

+

where

β

m and

α

mn are respectively the non-zero associated vector and associated tensor of recurrence.

(9)

According to the provisions of this definition, if we carry out calculations as have been carried out in the foregoing lines of this section, we can easily arrive at the following conclusions, which are being described in the form of corollaries.

COROLLARY (2.1): In a

R

+ generalized birecurrent space the following always holds

(2.17)

0

r nm r

r

R R

+ +

β

r

R N

+ mn

 

∂ − =

 

 



.

COROLLARY (2.2):

In a

R

+- generalized birecurrent space of the second kind the following identity always holds

(2.18) [ [ ] [

]

r r

r mn mn

r r

R

+

β R

+

α N R

+

N

+

 

+ +

 

 



( )

[

[ ] [ ] ]

r

m

R

n n

R

m r

R R

nm

β

+ +

− β

+ +

+  ∂

+

+

   

  



.

COROLLARY (2.3):

In a

R

+- generalized birecurrent space of the second kind the identity analogous to (4.15) is given by

(2.19)

[ ] [ ] ] ]

h h h

q t s r p

iqjk nm sqjk rt n ip m iqjh n m

x R

+ +

R

+

x Γ Γ

< >

xR

+ +

β −

  

[ ]

h iqjk nm

R

+

α

− 

[ [ ]

1 2

h h

r s t s r p

s sj

j r i nm rj n rt n ip m

R N R R x x

+ +

+

< >

 

= ∂ +  Γ + Γ  Γ +

 

  

[ ] [ ]

j i mn j n i m

R

+

α R

+ + < >

β

+ ∂  + ∂ 

.

Provided

Γ

ijkh be supposed to be first order recurrent with respect to associated vector of recurrence.

References

[1] Sen. R.N.: Finsler spaces of recurrent curvature Tensor (NS) 19, (1960) 291-299.

[2] Chaki, M.C. and Roy Chowdhary, A.N.: On Ricci recurrent spaces of second order, J. Ind.

Math. Soc. 2, 19, (1967).279-287.

[3] Mishra, R.S. and Pande, H.D.: Recurrent Finsler spaces, Jour. Ind. Math. Soc. 32, (1968) 17-20.

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[4] Ray, A.K.: On generalised 2-recurrent tensor in Riemannian spaces, Acad. Roy. Belg.

Bull. cl. Sci. (58), 5, (1972) 220-228.

[5] Pande, H.D. and Khan, T.A.: On generalized 2-recurrent Berwald’s curvature tensor field in a Finsler space, Acta Ciencia India 4(1), (1978) 56-59,

[6] Pande, P.N. and Misra, R.B.: Projective recurrent Finsler spaces, Publications Mathematica Debrecen, 28(3-4), (1981) 191-198.

[7] Pande, H.D. and Tewari, S.K.: Recurrent Finsler spaces, J. Nat. Acad. Math. Vol-II, (1991) 98-109.

[8] Meenakshy Thakur, C.K. Mishra and Gautam Lodhi, Decomposability of Projective Curvature Tensor in Recurrent Finsler Space (WR- Fn), IJCA, 2014, Vol. 100– No.19, 32-34.

References

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