Relations between the main scalars of a five-dimensional finsler space and its hypersurface

N/A
N/A
Protected

Share "Relations between the main scalars of a five-dimensional finsler space and its hypersurface"

Copied!
12
0
0

Full text

(1)

FINSLER SPACE AND ITS HYPERSURFACE

Department of Mathematics

A R T I C L E I N F O

INTRODUCTION

Let  us  consider  a  five-dimensional  Finsler  space

normalized supporting element, metric tensor and Cartan tensor are defined by

2 2

i i ij i j ijk k

respectively.

A hypersurface

4 of

M

5 may be represented parametrically by the equations  ordinate on

M

4 (Latin indices sum from 1 to 5, while Greek indices, except

matrix consisting of projection factors

i



vectors tangent to

4 at the point

form

i

i , where

X

 are the components of the vectors with respect to the coordinate system

To introduce a Finsler structure on

M

4, the supporting element  we may write

i i



Thus,

)

may  be  supposed  as  the  supporting  element  of  function

International Journal of Current Advanced Research

DOI: http://dx.doi.org/10.24327/ijcar.2017.

Article History:

August, 2017 Accepted 25th September, 2017   Published online 28th October, 2017

Key words:

Finsler space, hypersurface,   main scalars, Miron frame.

Copyright©2017 Anamika Rai and Tiwari S. K. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Corresponding author: Anamika Rai

Department of Mathematics, K S Saket P G College

Anamika Rai* and Tiwari S. K

of Mathematics, K S Saket P G College Ayodhya, Faizabad (India)

A B S T R A C T

Gauree Shanker, G. C. Chaubey and Vinay Pandey [1] studied a five

space  in  terms  of  scalars  with  the  help  of  ‘Miron  frame’  which  was  discussed  by  M.  Matsumoto and R. Miron [2]. On the other hand, the theory of hypersurface was discussed  in  detail  by  M.  Matsumoto  [3].  The  purpose  of  the  present  paper  is  to  obtain  relation  between  the  main  scalars  of  a  five-dimensional  Finsler  space  and  its  hypersurface.  For  terms and notations, we refer to Matsumoto [4].

dimensional  Finsler  space

5

5

L x y

whose  fundamental  metric  function  is  normalized supporting element, metric tensor and Cartan tensor are defined by

ij

i i ij i j ijk k



may be represented parametrically by the equations

i

i

)

, where

(Latin indices sum from 1 to 5, while Greek indices, except



take values 1 to 4). We assume that the  i

i

 



is of rank 4. Then

i

B u

may be regarded as four independent

  and a vector

i tangent to

M

4 at the point may be expressed uniquely in the  are the components of the vectors with respect to the coordinate system

, the supporting element

y

i is assumed to be tangent to

M

may  be  supposed  as  the  supporting  element  of

M

4  at  the  point u.  Denoting

International Journal of Current Advanced Research

http://dx.doi.org/10.24327/ijcar.2017.6511.0955

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Department of Mathematics, K S Saket P G College

DIMENSIONAL

Gauree Shanker, G. C. Chaubey and Vinay Pandey [1] studied a five- dimensional Finsler  terms  of  scalars  with  the  help  of  ‘Miron  frame’  which  was  discussed  by  M.  Matsumoto and R. Miron [2]. On the other hand, the theory of hypersurface was discussed  in  detail  by  M.  Matsumoto  [3].  The  purpose  of  the  present  paper  is  to  obtain  relation  dimensional  Finsler  space  and  its  hypersurface.  For

whose  fundamental  metric  function  is

.  The

, where

u

are the Gaussian co-take values 1 to 4). We assume that the

may be regarded as four independent

at the point may be expressed uniquely in the

are the components of the vectors with respect to the coordinate system

.

4

M

at a point u of

4, so that

(1)

.  Denoting

y

i  of  (1.1)  by

i

,  the

(2)

L x u y u v

(2)

gives rise to a Finsler metric on

M

4. Consequently, we get a four-dimensional Finsler space

4 4

where

L

is the induced metric function on

F

4.

The  induced  metric  function

yields

 



,  the  metric  tensor

2 2

  



and  the  Cartan  tensor



 

of  4

F

. Paying attention to

i

 



, from (2), we get

i i j i j k

i ij ijk





C B B B

(3)

At each point u of

F

4, a unit normal vector

i

is defined as

i j i j

ij ij



(4)

The inverse projection factors

i

of

B

i  is defined as

i

i ij

g g B

                             (5)

where

g

 is the inverse tensor of the metric tensor

 of

F

4.  From (5), it follows that

i i i

i i i

i i i i

i j j j

  

  

 



(6)

Let us deduce the following tensors from the Cartan tensor

ijk:

.

i j k i j k i j k

ijk ijk ijk



LC B B B

(7)

From (3), (6) and (7), we may write

j k

ijk i i

j k

ijk i i

j k

ijk i i

   

  

 



(8)

i i i

 

B

(9)

where

i

jk ijk

and

 

)

are called torsion vectors of

5 and

F

4 respectively.

Main Scalars of a five-dimensional Finsler space and its hypersurface

The Miron frame for a five-dimensional Finsler space is constructed by the unit vectors

i i

i

i

i

, where

i

i

L

is  the normalized supporting element,

i

i

C

is the normalized torsion vector (

C

is the length of the torsion vector

i);

n

i  is  constructed  by

ij i j

ij i j

ij i j

1

,  the  unit  vector

p

i  is  constructed  by

i j i j i j

ij ij ij

ij i j

and

q

i  is  constructed  by

ij i i

i i i i i i i j

ij ij ij ij



.

In the Miron frame,an arbitrary tensor

ji

)

is expressed in terms of scalar components as follows:

) )

i i

j j

(3)

where

1)i

i

2)i

i

3)i

i

4)i

i

5)i

q

i and the summation convection is applied to the indices

and

.   Let

C

 be the scalar components of

LC

ijk with respect to the Miron frame, i.e.

) ) )

ijk i j k

 

e

(10)

M. Matsumoto [4] showed that

1.

C

 are completely symmetric,

2.

1

,

3.

2

,

3

4

n

for

3

.

Therefore in five-dimensional Finsler space, we have

222 233 244 255

322 333 344 355

422 433 444 455

522 533 544 555



(11)

Thus putting

222 233 244 333 344

444 334 234 255 355

455 555 335 445 235

245 345

   

   

 



(12)

then, we have

223

224 225

 

     



Seventeen scalars H, I, K, J

,

M

,







N

,

M



are called the main scalars of a five-dimensional Finsler space.

The equation (10) may be written in expanded form as:

( ) ( )

( ) ( )

( ) ( )

)

ijk i j k ijk i j k ijk i j k i j k

ijk i j k i j k ijk i j k

ijk i j k ijk i j k i j k

 

   

   

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

)

ijk i j k ijk i j k ijk i j k ijk i j k

ijk i j k ijk i j k ijk i j k k j

ijk i j k k j ijk i j k k j

   

  

( )



ijk

i j k

k j

)}.

(13)

The hypersurface

4 of

F

5 is a four-dimensional Finsler space. The Moor frame for

4 is given by

, where

(



being  the  length  of  the  torsion  vector

  of

4),

n

  is  constructed  by

  

  

  

and

p

  is  constructed  by

  

  

  

  

(4)

( ) ( )

( ) ( )

( ) ( )

( )

n p

           

          

       

       

  

 

 

(14)

where



() denote the cyclic interchange of



, and summation and

,

,

 and

K

 are the  main scalars of

F

4. Transvecting (7) by

v

 and using (1), we get



. Therefore,

 and

M

 have

no component in the direction of

v

 (i.e. in the direction of

). Also,

M

 is symmetric. Therefore

and

M

 may be

written in the form

and



)

.   Thus, we have the following:

Proposition: Let

F

4 be the hypersurface of a five-dimensional Finsler space

F

5, then the tensor

and

M

 defined by (7),

are written as

and



m p

respectively.

From  (13)  and  (14),  the  torsion  vector

i  and

C

  are  represented  by

i

i  and

K m

respectively. The equation (19) and proposition lead to

1

i i i i

i

  

  

(15)

which yields

2 2 2 2 2

) .

(16)      Let us put

1 1 1 1

then,

i

i

i

i

i

.

(17)  Let us write the unit vectors

i,

i and

i as:

i i i i i

(18)

i i i i i

t B

             (19)

And

i

i

i

i

i

,

(20)  where abdtefgh

,

,

,

,

,

,

,

 are given by

   

   

   

   

       



and

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

   

   

 

(5)

From (21), we also have the relations:

   

   

   

   

   

and

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

 

 

 

 



(21)

Substituting (17), (18), (19) into (13) and using (3), we get

3 2 2 2 2 2 3

2 2 2 2 2 2 2 2

2 2 2 2 2 2

a e

   

     

        

         

2

2 2 2 2 2

2 2 2

2

I

      

 

       

         

      

  

2 2 2 2

2 2

2 2 2

2 2

a

   

       

            

       

2

2 2 2

2 2

af I

        

    

           

     

       

2 2 2 2

2 2 2 2

2 2

a b

         

     

       

  

2 2 2

2 2 2

m m p

         

             

            

2 2 2

2 2 2 2

2 2 2 2

    

       

      

(6)

2 2 2

2 2

aa g

          

           

              

     

2 2 2 2

2 2 2 2

2 2 2 2 2

2 2

eg

        

     

      

       

       

 

2

2 2

 

 

n n

     

            

              

        

2 2 2 2 2

2 2 2 2

2 2 2 2 2

2

bd K

         

       

      

     

2

2 2

2

2 2

bdd

     

  

    

             

            

        

  

2 2 2

2 2 2 2

2 2 2 2

2 2

f

      

       

     

        

   

2 2

2

3 2 2

2 2 2 2 2 2 2

f

  

            

           

          

      

2

2 2 2 2 2 2 2

2 2 2 2

3 2 2 2

 

b

         

           

            

    

2



2

b f

 2

(7)

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2

3 2 2

2 2 2 2

g

  

      

          

         

     

2 2 2

2 2 2 2 2 2 2

2 2 2 2

abg

              



        

           



fg

  

              

             

             

eb

                   

                

            

         

      

 



Comparing above equation with (14), we get

3 2 2 2 2 2 3 2

2 2 2 2 2 2 2 2

2 2 2 2 2 2

a

       

       

         

   

  

(23)

2 2 2 2 2

2 2 2 2

2 2 2 2

2 2

b a

         

       

     

    

2 2 2 2

2 2

N

    

            

              

         

(24)

2 2 2 2 2 2

2 2 2 2

2 2 2 2

2 2

d

         

       

     

    

2 2

a dg N

   

           

              

       



References

Related documents

It is probably a good practice for relocated students to begin to develop the “buffering identity function” to a capacity that will allow them to filter “racist information” and

The Effects of Early Enteral Nutrition Products on the Healing of Colo-Colonic Anastomosis 1 Department of General Surgery, Abant Izzet Baysal University Hospital, Bolu, Turkey 2

tion of hemoglobin was 16.6 gm/100 ml amid. there were 55 normoblasts per 100

The elevation of liver enzyme levels demand− ing therapy discontinuation occurred more fre− quently in patients over 65 years old who had been taking at least four other

In this study, there was an increase in the levels of uPA and MMP-3 in the endometrium of women with endometriosis, which can facilitate the adhesion of endometriosis tissue in

Although cortical and subcortical atrophy are con- sidered early markers of the disease, it is not known whether the absolute reduction in striatal volume is indicative for

This study was designed to detect the placental changes in hypertensive disorders of pregnancy and correlation with neonatal outcome.. The study was designed to analyze

For example, there is no explicit mention of how publicly-funded services will be affected by the introduction of the NHIS (e.g. would the insurance con- tribution made by