A Study on

A Study on ε − framed Metric Structure Manifold

AMIT MEWARI ^* , U.C. GAIROLA ^** and M. C. JOSHI ^***

* Department of Mathematics, Statistics & Computer Science, G.B. Pant University of Agriculture and Technology,

Pantnagar, Uttrarkhand, INDIA.

** Department of Mathematics,

Pauri Campus Pauri Garhwal, H.N.B. Garhwal, Srinagar Garhwal, Uttrarkhand, INDIA.

*** Department of Mathematics, D.S.B. Campus, Kumaon University,

Nainital, Uttrarkhand, INDIA.

(Received on: June 8, 2013)

ABSTRACT

In this paper, we have studied the ε − framed metric structure manifold and extend the results of K.K. Dube and N.K. Joshi a step forward. This manifold is very general manifold which in special cases reduces to framed metric manifold, almost r-contact metric manifold, almost contact metric manifold, almost r-Para Contact metric manifold almost contact metric manifold, almost Hermitian manifold, almost product Riemannian manifold.

Keywords: Framed metric manifold, Hermitian manifold, Riemannian manifold.

1. INTRODUCTION

In this paper, we consider (n=2m+r) dimension differentiable manifold M n of class C ^∞ , with tensor field F ≠ 0, and of type (1, 1), satisfying

)

2 (

k

k U

u I

F = ε − ⊗ (1.1) r -vector fields U ₁ , U ₂ , U ₃ … U _r and r- C ^∞ ,

1-forms u ¹ , u ² , u ³ … u ^r and a Riemannian metric g satisfying

= 0

FU k (1.2)

0

; ) (

=

= oF u

U u

k

k p p

k δ

Where

k

δ p ^is

Kronecker delta and k, p = 1, 2… r.

and X def F ( X ) ; For arbitrary vector field X.

∑

=

−

= ^r

k

k

k X u Y

u Y X g FY FX g

1

) ( ) ( )

, ( ) ,

( (1.3)

For all vector fields X, Y of M n , where k = 1, 2 … r and ε ² = 1 ^{7 . Then M} n is called an ε -framed metric manifold. Also in an ε -framed metric manifold, and we have g ( FX , Y ) = ε g ( X , FY ) .

In this structure, we have consider framed metric structure as ε =-1 throughout this paper.

This structure is very general which in special cases reduces to several known structures given below:

Structures/manifolds ε Framed metric Structure -1 Almost r-Contact Metric -1 Almost Contact metric -1 Almost r-Para Contact Metric 1 Almost Para Contact Metric 1

Almost Hermitian -1

Almost Product Riemannian 1 If there exists two vector fields X, Y which are tangent to manifold M n. The manifold M n satisfies above condition is called an ε -framed metric manifold. In this Manifold if we put X= U k in

∑

=

−

= ^r

k

k k k k

k FY g U Y u U u Y

FU g

1

) ( ) ( )

, ( ) , (

and using (1), we get g ( U _k , Y ) = u ^k ( Y ) ^. Similarly if we put Y= FY in (3) and using (1) and (2), we get,

) , ( ) ( ) , ( ) , (

) ( ) ( ) , ( ) ) (

, (

) ( ) ( ) , ( ) , (

1 1

2

k k

r

k

k k k

k

r

k

k k

U FX g Y u Y FX g FY X g

FY u X u FY X g Y U u I FX g

FY u X u FY X g Y F FX g

= +

−

=

⊗

−

−

=

∑

∑

=

=

ε

(1.5) Putting X =U m in (1.5) and making use of (1.2), we obtain,

0

) , (

) , ( ) ( ) , ( ) , (

=

= +

Y U g

U FU g Y u Y FX g Y FU g

m

k m k

m

(1.6) From (1.5) and (1.6),

0 ) , ( ) ,

( X FY + g FX Y =

g (1.7)

Let us define 2-form F ′ as

) , ( ) ,

( X Y g FX Y

F ′ = (1.8) In this continuation, we have deduced the results in form of two theorems.

2. MAIN RESULTS

Theorem 1: A ε − framed metric structure manifold is not unique. If µ be a non- singular vector valued function of M n , Let us put,

 

 



=

=

′ =

k k

k k

U V iii

o u V ii

Fo F o i

µ

µ µ µ

) ) )

(2.1)

Then { ^{F ′ , V k , V k} } ^{gives a ε −} framed metric structure on M n.

Proof: we have µ o F ′ = Fo µ

On post multiplying (2.1) (i) by F′ and making use of (1.1) and (2.1); we get,

µ µ

µ ^{o F} ′ ² = ^{Fo o F} ′ = ^{F 2 o} Now,

) ) ( (

) )

( (

) (

2 2 2

k k

k k

k k

V u

F o

o U u

F o

o U u I F

o

µ µ

ε µ

µ µ

µ ε µ

µ ε

µ

−

′ =

−

′ =

⊗

−

′ =

From (2.1) (ii), we get,

) 3 . 2 ( )

(

) 2 . 2 ( )

(

2 2

k k n

k k

V V I F

V V F

o

⊗

−

′ =

⊗

−

′ = ε

µ ε µ

Also from (2.1) (i) and (iii), we have

= 0

′ V _k = Fo V _k F

o µ

µ ^{. Thus,}

FU k =0, k = 1, 2, …, r. (2.4) Again, V ^k oF = u ^k o µ o F ′ = u ^k oF = 0 by (1.1).

Thus, V ^k o F ′ = 0 (2.5) Further,

k p p k oV

u = δ ,k, p =1,2,…, r (2.6) By virtue of equation (2.3), (2.4), (2.5) and (2.6).we conclude that

{ ^{F ′ , V k , V k} } gives an ε − framed metric structure on M n.

Theorem 2: The necessary and sufficient condition that, M n be an ε − framed metric structure manifold is that it possesses a tangent bundle π _m of dimension m, tangent bundle

~

π m conjugate to π _m and the product set π _r ( R ^r ) of ordered r – tuples of real numbers such that,

ϕ π π π π π

π m ∩ m = m ∩ m = m ∩ r =

~

~

and they span together a tangent bundle of dimension n =(2m+r). Projection L, M, N on

π m ^, π ^~ m ^and π _r are given by,

 



 



⊗

=

−

=

+

−

−

−

k k

m u U

I F N iii

iF F def M ii

iF F def L i

ε

2 2 2

) 2 )

2 )

Proof: Suppose that, M n admits a ε − ^framed metric structure. Hence; corresponding to eigenvalues i [ ] ^{2 , let} P k ^; k = 1, 2, …, r is n linearly independent eigenvalues. Let Q _k be eigenvectors Conjugate to P _k .Further, there is r – linearly independent vector field U _k . Thus, we have

( ^{, , 0 , )}

, 0

, 0

Scalars are

c b a

c U c

b Q b

a P a

k k k

k k k

k k k

k k k

=

⇒

=

⇒

=

⇒

Now, if

) 8 . 2 ( , 0

) 7 . 2 ( 0

= +

+

= +

+

k k k k k k

k k k k k k

U c Q b P a

U c Q b P

a

In view of equation (2.1.2) and we know the fact that P , _k Q _k are eigenvectors corresponding to eigenvalues i and –i respectively; we have

) 9 . 2 ( ,

= 0

− k

k k

k P b Q

a

k = 1, 2, …, n.

Baring (2.9) again and using the same fact that P _k Q _k are eigenvectors corresponding i and –i, we get,

) 10 . 2 (

= 0

+ k

k k

k P b Q

a

Thus, from (2.9) and (2.10); we have

. ,...

2 , 1 ,

0 k n

b

a ^k = ^k = =

Thus, form (2.7); it follows that

= 0 c k

Thus; { ^{P k , , Q k , , U k } are linearly}

independent set.

From eqn. (3.1.8) we can easily show that, (i) LP _{K ,} = P _K , (ii) LQ _K = 0 , (iii) LU _K = 0 . … (2.11) (i) MP _K = 0 , (ii) MQ _K = Q _{K ;} (iii) MU _K = 0 . … (2.12) (i) NP _K = 0 , (ii) NQ _K = 0 ; (iii) NU _K = U _K . …(2.13)

Thus, there exists a tangent bundle π m of dimension and a tangent bundle

~

π m

Conjugate to π _m and the product space π _r

or, ordered r-tuples of real numbers such that ϕ π π π π π

π m ∩ m = m ∩ r = m ∩ r =

~

~

and

r m

m π π

π ∪ ^~ ∪ gives a tangent bundle of dimension (2m+r), projections on π _m ^, π ^~ m

and π _r being L, M and N respectively.

Suppose, conversely that in M _n these exists a tangent bundle π _m ^of dimension m, π _m conjugate to

~

π m ^and

product set π _r such that they are mutually disjoint and span together a tangent bundle of dimension n.

Let P _k be m linearly independent vectors in π _m ^{, Qk in} π ^~ m conjugate to Pk

and Uk be r- linearly independent vectors in product set π _r . Suppose { P k , ^, Q k , ^, U k ^}

span a tangent bundle of dimension (2m+r).

Define the inverse set { ^{p k , q k , u k } as}

n K K K K K

K P q Q u U I

p ⊗ + ⊗ + ⊗ = (2.14)

Let, us now put,

{ K ^}

K K

K P q Q

p def

F ⊗ − ⊗ (2.15)

Thus, we have,

{ k }

K k

K P q Q

p i

F ² = ⊗ − ⊗ (2.16)

In view of equation (2.15), the

above equation takes form

{ ^{p k P k q K Q K }}

F ² = − ⊗ + ⊗ . This by

virtue of (2.14) takes the form

) (

,

2 2

k k n

k k n

U u I F

or

U u I F

⊗

−

=

⊗ +

−

= ε

Thus, M n admits a ε − framed metric structure.

Remark 1: In Theorem 1and 2, if we take

− 1

ε = .We gets the result of Nivas Ram and Rajesh Singh ⁶ .

REFERENCES

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