Cmjv04i03p0197 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.4 (3), 197-201 (2013)

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 Mn of class C ∞ , with tensor field F ≠ 0, and of type (1, 1), satisfying

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

FU k = 0

(1.2) k

u k (U p ) = δ ; p

F = ε (I − u ⊗ U k ) 2

(1.1)

u oF = 0

∞

Where δ is

r -vector fields U 1 , U 2 , U 3 … U r and r- C ,

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Amit Mewari, et al., J. Comp. & Math. Sci. Vol.4 (3), 197-201 (2013) r

Kronecker delta and k, p = 1, 2… r.

g ( FX , F 2Y ) = g ( X , FY ) − ∑ u k ( X )u k ( FY ) k =1

and X def F ( X ) ; For arbitrary vector

g ( FX , ε ( I − u ⊗ U k )Y ) = g ( X , FY ) − ∑ u k ( X )u k ( FY ) k

field X.

k =1

g ( X , FY ) + g ( FX , Y ) = u k (Y ) g ( FX , U k ) r

g ( FX , FY ) = g ( X , Y ) − ∑ u k ( X )u k (Y ) (1.3) k =1

For all vector fields X, Y of Mn, where k = 1, 2 … r and ε 2 = 1 7. Then Mn 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 Framed metric Structure Almost r-Contact Metric Almost Contact metric Almost r-Para Contact Metric Almost Para Contact Metric Almost Hermitian Almost Product Riemannian

ε -1 -1 -1 1 1 -1 1

If there exists two vector fields X, Y which are tangent to manifold Mn. The manifold Mn satisfies above condition is called an ε -framed metric manifold. In this Manifold if we put X= Uk in r

g ( FU k , FY ) = g (U k , Y ) − ∑ u k (U k )u k (Y ) k =1

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

(1.5) Putting X =Um in (1.5) and making use of (1.2), we obtain, g ( FU m , Y ) + g ( FX , Y ) = u k (Y ) g ( FU m , U k ) g (U m , Y ) = 0

(1.6)

From (1.5) and (1.6),

g ( X , FY ) + g ( FX , Y ) = 0

(1.7)

Let us define 2-form F ′ as

F ′( X , Y ) = g ( FX , Y )

(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 nonsingular vector valued function of Mn, Let us put,

i ) µ o F ′ = Foµ  k k ii ) V = u oµ iii ) µV = U k k 

{

Then F ′, V k , V k structure on Mn.

(2.1)

} gives a ε −framed metric

Proof: we have µ o F ′ = Fo µ

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Amit Mewari, et al., J. Comp. & Math. Sci. Vol.4 (3), 197-201 (2013)

199

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

π m ∩ π m = π m ∩ π m = π m ∩ π r = ϕ and

µ o F ′ 2 = Fo µ o F ′ = F 2 o µ

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

Now,

π m , π m and π r are given by,

µ o F ′ 2 = ε ( I − u k ⊗ U k ) oµ µ o F ′ 2 = ε ( µ − u k ( µ ) U k oµ ) µ o F ′ 2 = ε ( µ − u k ( µ ) Vk ) From (2.1) (ii), we get,

µ o F ′ 2 = ε ( µ − V k ⊗ Vk )

( 2 .2 )

F ′ 2 = ε ( I n − V k ⊗ Vk )

( 2 . 3)

Also from (2.1) (i) and (iii), we have µ o F ′Vk = Foµ Vk = 0 . Thus,

FU k =0, k = 1, 2, …, r.

i ) 2 L def − F 2 − iF   2 ii ) 2 M def − F + iF  2 k iii ) N = F − εI m = u ⊗ U k Proof: Suppose that, Mn admits a ε − framed metric structure. Hence; corresponding to eigenvalues i [2], let Pk ; k = 1, 2, …, r is n linearly independent eigenvalues. Let Qk be

(2.4)

eigenvectors Conjugate to Pk .Further, there

Again, V k oF = u k oµ oF ′ = u k oF = 0 by (1.1). Thus, V k oF ′ = 0 (2.5)

is r – linearly independent vector field U k . Thus, we have

Further, u k oV p = δ ,k, p =1,2,…, r p

(2.6)

By virtue of equation (2.3), (2.4), (2.5) and (2.6).we conclude that k gives an ε −framed metric F ′, V k , V structure on Mn.

{

}

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

a k Pk ⇒ a k = 0, b k Qk ⇒ b k = 0, c k U k ⇒ c k = 0,

, b k , c k are Scalars )

Now, if a k Pk + b k Q k + c k U a k Pk

= 0

+ bk Qk + ckU

= 0,

( 2 .7 ) ( 2 .8 )

In view of equation (2.1.2) and we

bundle π m conjugate to π m and the product

know the fact that Pk , Qk are eigenvectors corresponding to eigenvalues i and –i respectively; we have

set π r ( R r ) of ordered r – tuples of real numbers such that,

a k Pk − b k Q k = 0 ,

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( 2 .9 )

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k = 1, 2, …, n.

a k = b k = 0,

Baring (2.9) again and using the same fact that Pk Qk are eigenvectors corresponding i and –i, we get,

Thus, form (2.7); it follows that

a k Pk + b k Q k = 0

( 2 . 10 )

k = 1,2,.........n.

ck = 0

{P , Q

Thus;

, Uk }

are

linearly

independent set. From eqn. (3.1.8) we can easily show that,

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

LPK , = PK ,

(ii) LQK = 0,

(iii)

LU K = 0.

… (2.11)

(i)

MPK = 0,

(ii) MQ K = Q K ;

(iii)

MU K = 0.

… (2.12)

(i)

NPK = 0,

(ii) NQK = 0;

(iii) NU K = U K .

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 ~

πm ∩πm = πm ∩πr = πm∩πr = ϕ

and

…(2.13)

and Uk be r- linearly independent vectors in product set π r . Suppose Pk , , Qk , , U k }

{

span a tangent bundle of dimension (2m+r). Define the inverse set p k , q k , u k } as

{

p K ⊗ PK + q K ⊗ QK + u K ⊗ U K = I n (2.14)

π m ∪ π m ∪ π r gives a tangent bundle of ~

dimension (2m+r), projections on π m , π m and π r being L, M and N respectively.

}

(2.15)

}

(2.16)

{

F 2 = i p K ⊗ P k − q K ⊗ Qk

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 Pk be m linearly independent ~

{

F def p K ⊗ PK − q K ⊗ QK Thus, we have,

Suppose, conversely that in M n these exists a tangent bundle π m

Let, us now put,

vectors in π m , Qk in π m conjugate to Pk

In view of equation (2.15), the equation takes form 2 k K F = − p ⊗ Pk + q ⊗ Q K } . This by virtue of (2.14) takes the form above

{

F 2 = −I n + u k ⊗ U k or, F 2 = ε (I n − u k ⊗ U k )

Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)

Amit Mewari, et al., J. Comp. & Math. Sci. Vol.4 (3), 197-201 (2013)

Thus, Mn admits a structure.

ε −framed metric 5.

Remark 1: In Theorem 1and 2, if we take ε = −1.We gets the result of Nivas Ram and Rajesh Singh6. 6. REFERENCES 1. Joshi, N. K., Dube, K. K., Semi invariant of a ε − framed metric structure manifold, Acta Cienecia Indica, Vol.29, No. 1,139 (2003). 2. Tripathi, M.M., Singh, K. D., Almost Semi invariant submanifold of a ε − framed Metric structure manifold, Acta Cienecia Indica, Vol.29, No. 413-426 (1996). 3. Nivas Ram, Rajesh Singh, On Almost rContact Structure Manifolds, Vol.XXI, No.3, 797-803 (1988). 4. Gupta V.C., Prasad C.S., Almost Para r-

7. 8.

10.

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contact structures manifolds, Demonstration Math. XIX(4), 1-15 (1986). Kobayashi M., Nomizu K., Foundation of Differential Geometry, Vol.(I, II), Interscience Publishers, New York (1986). Sinha, B.B., An Introduction to modern Differential Geometry, Kalyani Publishers, New Delhi (1982). Matsuhima Y., Differentiable Manifold, Marcell Dekker, Inc., New York (1972). Kobayashi, S. and Nomizu, K., Foundation of Differentiable Geometry, Vol (I, II), Inter Science Publishers, New York (1969). Hicks, N.J., Notes on Differentiable Geometry; Van Nostrand, Press, New York (1969). Sasaki, S., On Differentiable Manifolds with certain structures which are closely related to Almost Contact Structure I, Tohoku Math. J., 12, 459-476 (1960).

Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)