Beta −Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces

International J.Math. Combin. Vol.4(2017), 46-59

β−Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces O.P.Pandey and H.S.Shukla (Department of Mathematics & Statistics, DDU Gorakhpur University,Gorakhpur, India) E-mail: oppandey1988@gmail.com, profhsshuklagkp@rediffmail.com

Abstract: We have considered the β−change of Finsler metric L given by L = f (L, β) where f is any positively homogeneous function of degree one in L and β. Here β = bi (x, y)y i , in which bi are components of a covariant h-vector in Finsler space F n with metric L. We have obtained that due to this change of Finsler metric, the imbedding class of their tangent Riemannian space is increased at the most by two.

Key Words: β−Change of Finsler metric, h-vector, imbedding class.

AMS(2010): 53B20, 53B28, 53B40, 53B18. §1. Introduction Let (M n , L) be an n-dimensional Finsler space on a differentiable manifold M n , equipped with the fundamental function L(x, y). In 1971, Matsumoto [2] introduced the transformation of Finsler metric given by L(x, y) = L(x, y) + β(x, y), (1.1) 2

L (x, y) = L2 (x, y) + β 2 (x, y),

(1.2)

where β(x, y) = bi (x)y i is a one-form on M n . He has proved the following.

Theorem A. Let (M n , L) be a locally Minkowskian n-space obtained from a locally Minkowskian n-space (M n , L) by the change (1.1). If the tangent Riemannian n-space (Mxn , gx ) to (M n , L) is of imbedding class r, then the tangent Riemannian n-space (Mxn , g x ) to (M n , L) is of imbedding class at most r + 2.

Theorem B. Let (M n , L) be a locally Minkowskian n-space obtained from a locally Minkowskian n-space (M n , L) by the change (1.2). If the tangent Riemannian n-space (Mxn , gx ) to (M n , L) is of imbedding class r, then the tangent Riemannian n-space (Mxn , g x ) to (M n , L) is of imbedding class at most r + 1.

Theorem B is included in theorem A due to the phrase “at most ”.

In [6] Singh, Prasad and Kumari Bindu have proved that the theorem A is valid for Kropina 1 Received

April 18, 2017, Accepted November 8, 2017.

β−Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces

47

change of Finsler metric given by L(x, y) =

L2 (x, y) . β(x, y)

In [3], Prasad, Shukla and Pandey have proved that the theorem A is also valid for exponential change of Finsler metric given by L(x, y) = Leβ/L .

Recently Prasad and Kumari Bindu [5] have proved that the theorem A is valid for β−change [7] given by L(x, y) = f (L, β), where f is any positively homogeneous function of degree one in L, β and β is one-form.

In all these works it has been assumed that bi (x) in β is a function of positional coordinate only.

The concept of h−vector has been introduced by H.Izumi. The covariant vector field ∂bi bi (x, y) is said to be h−vector if ∂y j is proportional to angular metric tensor.

In 1990, Prasad, Shukla and Singh [4] have proved that the theorem A is valid for the transformation (1.1) in which bi in β is h−vector.

All the above β−changes of Finsler metric encourage the authors to check whether the theorem A is valid for any change of Finsler metric by h−vector.

In this paper we have proved that the theorem A is valid for the β−change of Finsler metric given by L(x, y) = f (L, β), (1.3) where f is positively homogeneous function of degree one in L, β and β(x, y) = bi (x, y)y i .

(1.4)

Here bi (x, y) are components of a covariant h−vector satisfying ∂bi = ρhij , ∂y j

(1.5)

where ρ is any scalar function of x, y and hij are components of angular metric tensor. The homogeneity of f gives Lf1 + βf2 = f, (1.6) where the subscripts 1 and 2 denote the partial derivatives with respect to L and β respectively.

Differentiating (1.6) with respect to L and β respectively, we get Lf11 + βf12 = 0 and Lf12 + βf22 = 0.

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O.P.Pandey and H.S.Shukla

Hence, we have

f11 f12 f22 =− = 2 2 β βL L

which gives f11 = β 2 ω,

f22 = L2 ω,

f12 = −βLω,

(1.7)

where Weierstrass function ω is positively homogeneous function of degree −3 in L and β.

Therefore Lω1 + βω2 + 3ω = 0.

(1.8) Again ω1 and ω2 are positively homogeneous function of degree - 4 in L and β, so Lω11 + βω12 + 4ω1 = 0 and Lω21 + βω22 + 4ω2 = 0.

(1.9)

Throughout the paper we frequently use equation (1.6) to (1.9) without quoting them.

§2. An h−Vector Let bi (x, y) be components of a covariant vector in the Finsler space (M n , L). It is called an h−vector if there exists a scalar function ρ such that ∂bi = ρhij , ∂y j

(2.1)

where hij are components of angular metric tensor given by hij = gij − li lj = L

∂2L . ∂y i ∂y j

Differentiating (2.1) with respect to y k , we get ∂˙j ∂˙k bi = (∂˙k ρ)hij + ρL−1 {L2 ∂˙i ∂˙j ∂˙k L + hij lk }, where ∂˙i stands for

∂ ∂y i .

The skew-symmetric part of the above equation in j and k gives (∂˙k ρ + ρL−1 lk )hij − (∂˙j ρ + ρL−1 lj )hik = 0.

Contracting this equation by g ij , we get (n − 2)[∂˙k ρ + ρL−1 lk ] = 0, which for n > 2, gives

ρ ∂˙k ρ = − lk , L

(2.2)

where we have used the fact that ρ is positively homogeneous function of degree −1 in y i , i.e.,

β−Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces

∂ρ j ∂y j y

49

= −ρ.

We shall frequently use equation (2.2) without quoting it in the next article.

§3. Fundamental Quantities of (M n , L)

To find the relation between fundamental quantities of (M n , L) and (M n , L), we use the following results ∂˙i β = bi , ∂˙i L = li , ∂˙j li = L−1 hij .

(3.1) The successive differentiation of (1.3) with respect to y i and y j give

hij =

l i = f1 li + f2 b i ,

(3.2)

fp hij + f L2 wmi mj , L

(3.3)

where p = f1 + Lf2 ρ, mi = bi −

β li .

L

The quantities corresponding to (M n , L) will be denoted by putting bar on the top of those quantities.

From (3.2) and (3.3) we get the following relations between metric tensors of (M n , L) and (M n , L) g ij

=

fp gij − L−1 {β(f1 f2 − f βLω) + Lρf f2 }li lj L +(f L2 ω + f22 )bi bj + (f1 f2 − f βLω)(li bj + lj bi ).

(3.4)

The contravariant components of the metric tensor of (M n , L) will be obtained from (3.4) as follows: L ij Lv L4 ω i j L2 u gij = g + 3 li lj − b b − 2 (li bj + lj bi ), (3.5) fp f pt f pt f pt where, we put bi = g ij bj , li = g ij lj , b2 = g ij bi bj and u v

= f1 f2 − f βLω + Lρf22 ,

= (f1 f2 − f βLω)(f β + △f2 L2 ) + Lρf2 {f (f + L2 ρf2 ) +L2 △(f22 + f L2 ω)}

and t = f1 + L3 ω△ + Lf2 ρ,

△ = b2 −

β2 .

L2

(3.6)

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O.P.Pandey and H.S.Shukla

Putting q = f1 f2 − f βLω + Lρ(f22 + f L2 ω), s = 3f2 ω + f ω2 , we find that (a) (b) (c) (d) (e) (f) (g)

(a) (b) (c)

f ∂˙i f = li + f2 mi L ∂˙i f1 = −βLωmi ∂˙i f2 = L2 ωmi ∂˙i p = −Lω(β − ρL2 )mi 3ω ∂˙i ω = − li + ω2 mi L 2 ˙ ∂i b = −2C..i + 2ρmi 2 ∂˙i △ = −2C..i − 2 (β − ρL2 )mi , L

(3.7)

∂˙i q = −(β − ρL2 )sLmi ∂˙i t = −2L3 ωC..i + [L3 △ω2 − 3(β − ρL2 )Lω]mi 3s ∂˙i s = − li + (4f2 ω2 + 3ω 2 L2 + f ω22 )mi L

(3.8)

where “.” denotes the contraction with bi , viz. C..i = Cjki bj bk .

Differentiating (3.4) with respect to y k and using (d that mi li = 0, where mi = g ij mj = bi −

mi mi = △ = mi b i ,

hij mj = hij bj = mi ,

(3.10)

β i Ll .

i

To find C jk = g ih C jhk we use (3.5), (3.9), (3.10) and get i

C jk

=

q sL3 L (hjk mi + hij mk + hik mj ) + mj mk mi − C.jk ni 2f p 2f p ft 4 Lq△ 2Lq + L △s − 2 hjk ni − mj mk n i , (3.11) 2f pt 2f 2 pt

i Cjk +

where ni = f L2 ωbi + uli .

Corresponding to the vectors with components ni and mi , we have the following: Cijk mi = C.jk ,

Cijk ni = f L2 ωC.jk ,

mi ni = f L2 ω△.

(3.12)

To find the v-curvature tensor of (M n , L) with respect to Cartan’s connection, we use the following: h Cij hhk = Cijk , hik hkj = hij , hij ni = f L2 ωmj .

(3.13) The v-curvature tensors S hijk of (M n , L) is defined as r

r

S hijk = C hk Chjr − C hj C ikr .

(3.14)

From (3.9)–(3.14), we get the following relation between v-curvature tensors of (M n , L)

β−Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces

51

and (M n , L): S hijk =

fp Shijk + dhj dik − dhk dij + Ehk Eij − Ehj Eik , L

(3.15)

where

P =L

f pω t

dij = P C.ij − Qhij + Rmi mj ,

(3.16)

Eij = Shij + T mi mj ,

(3.17)

1/2

S=

,

Q=

pq √ , 2L2 f pωt

q √ , 2L2 f ω

T =

R=

L(2ωq − sp) √ , 2 f ωpt

L(sp − ωq) √ .

2p f ω

§4. Imbedding Class Numbers

The tangent vector space Mxn to M n at every point x is considered as the Riemannian nspace (Mxn , gx ) with the Riemannian metric gx = gij (x, y)dy i dy j . Then the components of the Cartan’s tensor are the Christoffel symbols associated with gx : i = Cjk

1 ih ˙ g (∂k gjh + ∂˙j ghk − ∂˙h gjk ).

2

i Thus Cjk defines the components of the Riemannian connection on Mxn and v-covariant derivative, say h Xi |j = ∂˙j Xi − Xh Cij i is the covariant derivative of covariant vector Xi with respect to Riemannian connection Cjk on n n Mx . It is observed that the v-curvature tensor Shijk of (M , L) is the Riemannian Christoffel curvature tensor of the Riemannian space (M n , gx ) at a point x. The space (M n , gx ) equipped with such a Riemannian connection is called the tangent Riemannian n-space [2].

It is well known [1] that any Riemannian n-space V n can be imbedded isometrically in a Euclidean space of dimension n(n+1) . If n + r is the lowest dimension of the Euclidean space 2 n in which V is imbedded isometrically, then the integer r is called the imbedding class number of V n . The fundamental theorem of isometric imbedding ([1] page 190) is that the tangent Riemannian n-space (Mxn , gx ) is locally imbedded isometrically in a Euclidean (n + r)−space if and only if there exist r−number ǫP = ±1, r−symmetric tensors H(P )ij and r(r−1) covariant 2 vector fields H(P,Q)i = −H(Q,P )i ; P, Q = 1, 2, · · · , r, satisfying the Gauss equations Shijk =

X P

ǫP {H(P )hj H(P )ik − H(P )ij H(P )hk },

(4.1)

The Codazzi equations H(P )ij |k − H(P )ik |j =

X Q

ǫQ {H(Q)ij H(Q,P )k − H(Q)ik H(Q,P )j },

(4.2)

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O.P.Pandey and H.S.Shukla

and the Ricci-Kühne equations H(P,Q)i |j − H(P,Q)j |i

+ +

X

R hk

g

ǫR {H(R,P )i H(R,Q)j − H(R,P )j H(R,Q)i } {H(P )hi H(Q)kj − H(P )hj H(Q)ki } = 0.

(4.3)

The numbers ǫP = ±1 are the indicators of unit normal vector NP to M n and H(P )ij are the second fundamental tensors of M n with respect to the normals NP . Thus if gx is assumed to be positive definite, there exists a Cartesian coordinate system (z i , up ) of the enveloping Euclidean space E n+r such that ds2 in E n+r is expressed as ds2 =

X X (dz i )2 + ǫp (dup )2 . i

p

§5. Proof of Theorem A In order to prove the theorem A, we put r

fp ǫP = ǫP , P = 1, 2, · · · , r H(P )ij , L = dij , ǫr+1 = 1

(a)

H (P )ij =

(b)

H (r+1)ij

(c)

H (r+2)ij = Eij ,

ǫr+2 = −1.

(5.1)

Then it follows from (3.15) and (4.1) that S hijk =

r+2 X

λ=1

ǫλ {H (λ)hj H (λ)ik − H (λ)hk H (λ)ij },

which is the Gauss equation of (Mxn , gx ).

Moreover, to verify Codazzi and Ricci Kühne equation of (Mxn , g x ), we put (a) (b) (c) (d)

H (P,Q)i = −H (Q,P )i = H(P,Q)i , P, Q = 1, 2, , · · · , r √ L Lω H (P,r+1)i = −H (r+1,P )i = √ H(P ).i , P = 1, 2, · · · , r t H (P,r+2)i = −H (r+2,P )i = 0, P = 1, 2, · · · , r. sp − 2qω √ mi .

H (r+1,r+2)i = −H (r+2,r+1)i = 2f ω pt

(5.2)

The Codazzi equations of (Mxn , g x ) consists of the following three equations: (a) H (P )ij kk − H (P )ik kj =

X Q

ǫQ {H (Q)ij H (Q,P )k − H (Q)ik H (Q,P )j }

+ ǫr+1 {H (r+1)ij H (r+1,P )k − H (r+1)ik H (r+1,P )j } + ǫr+2 {H (r+2)ij H (r+2,P )k − H (r+2)ik H (r+2,P )k }

(5.3)

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β−Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces

(b)

H (r+1)ij kk − H (r+1)ik kj

=

X

ǫQ {H (Q)ij H (Q,r+1)k − H (Q)ik H (Q,r+1)j }

X

ǫQ {H (Q)ij H (Q,r+2)k − H (Q)ik H (Q,r+2)j }

Q

+ǫr+2 {H (r+2)ij H (r+2,r+1)k − H (r+2)ik H (r+2,r+1)j },

(c)

H (r+2)ij kk − H (r+2)ik kj

=

Q

+ǫr+1 {H (r+1)ij H (r+1,r+2)k − H (r+1)ik H (r+1,r+2)j }. where ki denotes v-covariant derivative in (M n , L), i.e. covariant derivative in tangent Riemani nian n-space (Mxn , g x ) with respect to its Christoffel symbols C jk . Thus h Xi kj = ∂˙j Xi − Xh C ij .

To prove these equations we note that for any symmetric tensor Xij satisfying Xij li = Xij l = 0, we have from (3.11), j

q (hik X.j − hij X.k ) 2f t L3 ω q + (C.ik X.j − C.ij X.k ) − (Xij mk − Xik mj ) t 2f p L3 (2qω − sp) + (X.j mk − X.k mj ) mi .

(5.4) 2f pt

Xij kk − Xik kj = Xij |k − Xik |j −

Also if X is any scalar function, then Xkj = X|j = ∂˙j X. Verification of (5.3)(a) In view of (5.1) and (5.2), equation (5.3)a is equivalent to ! ! r

fp fp

H(P )ij − H(P )ik L L k j r √ fp X L Lω = . ǫQ {H(Q)ij H(Q,P )k − H(Q)ik H(Q,P )j } − √ {dij H(P ).k − dik H(P ).j }.

L t Q r

Since

q

fp L

q

fp ˙ = = ∂k L k

√q 2 f Lp

mk , applying formula (5.4) for H(P )ij , we get

! ! r r

fp fp fp

H H {H(P )ij |k − H(P )ik |j } − = L (P )ij L (P )ik L k j r r q fp L3 ω f p − {hik H(P ).j − hij H(P ).k } + {C.ik H(p).j − C.ij H(p).k } 2f t L t L √ L2 L(2qω − sp) √ + {H(P ).j mk − H(P ).k mj }mi .

2t f p r

Substituting the values of

q

fp L H(P )ij

(5.5)

(5.6)

q

fp H − (P )ik from (5.6) and the values L k

j

of dij from (3.16) in (5.5) we find that equation (5.5) is identically satisfied due to equation

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O.P.Pandey and H.S.Shukla

(4.2).

Verification of (5.3)(b) In view of (5.1) and (5.2), equation (5.3)b is equivalent to dij kk − dik kj = L

r

f ωp X ǫQ {H(Q)ij H(Q).k − H(Q)ik H(Q).j } t Q

sp − 2qω √ {Eij mk − Eik mj }. + 2f ω pt

(5.7)

To verify (5.7), we note that C.ij |k − C.ik |j = −bh Shijk

(5.8)

hij |k − hik |j = L−1 (hij lk − hik lj ),

1 β mi |k = −C.ik − − ρ hik − li mk .

L2 L

(5.9)

∂˙k (f ωp) = −2L−1 f ωplk + (qω + f pω2 )mk .

(5.10) (5.11)

Contracting (3.16) with bi and using (3.10), we find that d.j = L

r

f ωp q(2L3 ω△ − p) − L3 △sp √ mj .

C..j + t 2L2 f ωpt

(5.12)

Applying formula (5.4) for dij and substituting the values of d.j from (5.12) and dij from (3.16), we get dij kk − dik k= j

√ Lq f ωp (hik C..j − hij C..k ) 2f t3/2 L4 ω(2qω − sp) + √ (C..j mk − C..k mj )mi 2 f ωp.t3/2 √ L4 ω f ωp + (C.ik C..j − C.ij C..k ) t3/2 L4 ω△(3qω − sp) √ + (C.ik mj − C.ij mk ) 2 f ωp.t3/2 Lq△(3qω − sp) √ − (hik mj − hij mk ).

4f f ωp.t3/2

dij |k − dik |j −

(5.13)

From (3.16), we obtain dij |k − dik |j = P (C.ij |k − C.ik |j ) − Q(hij |k − hik |j ) +R(mi |k mj + mj |k mi − mi |j mk − mk |j mi ) +(∂˙k P )C.ij − (∂˙j P )C.ik − (∂˙k Q)hij + (∂˙j Q)hik )

+(∂˙k R)mi mj − (∂˙j R)mi mk ).

(5.14)

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β−Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces

Since, √

L4 ω f ωp Lf p{pω2 + 3Lω 2 (β − ρL2 )} √ ∂˙k P = C + ..k 3/2 t 2 f ωp.t3/2

Lqω + √ mk , 2 f ωpt Lpqω pq √ ∂˙k Q = √ C..k − lk 3/2 3 2 f ωp.t 2L f ωpt (β − ρL2 )(qω + sp) pq(qω + f pω2 ) √ √ mk mk − − 2L f ωpt 4L2 (f ωp)3/2 t pq{3ω(β − ρL2 ) − L2 △ω2 } √ + mk 4L f ωp t3/2

(5.15)

and 2qω − sp L4 ω(2qω − sp) √ C..k − √ lk + term containing mk , ∂˙k R = 2 f ωp.t3/2 2 f ωpt where we have used the equations (3.6), (3.7) and (3.8).

From equations (5.8)–(5.15), we have r

f ωp (−bh Shijk ) t L4 ω△(3qω − sp) √ + (C.ij mk − C.ik mj ) 2 f ωp.t3/2 √ L4 ω f ωp + (C.ij C..k − C.ik C..j ) t3/2 Lωpq + √ (hik C..j − hij C..k ) 2 f ωp.t3/2 pq[qωt + f (L3 ω△ + t){3Lω 2 (β − ρL2 ) + pω2 }] + × 4L2 (f ωpt)3/2 L4 ω(2qω − sp) √ (hij mk − hik mj ) + (C..k mj − C..j mk ) mi .

2 f ωp.t3/2

dij |k − dik |j = L

(5.16)

Substituting the value of dij |k − dik |j from (5.16) in (5.13), then value of dij kk − dik kj thus obtained in (5.7), and using equations (4.1) and (3.17), it follows that equation (5.7) holds identically.

Verification of (5.3)(c) In view of (5.1) and (5.2), equation (5.3)c is equivalent to Eij kk − Eik kj =

sp − 2qω √ (dij mk − dik mj ).

2f ω pt

(5.17)

Contracting (3.17) by bi and using equation (3.10), we find that E.j =

pq + L3 △(sp − qω) √ mj .

2L2 p f ω

(5.18)

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O.P.Pandey and H.S.Shukla

Applying formula (5.4) for Eij and substituting the value of E.j from (5.18) and the value of Eij from (3.17), we get qL△(sp − 2qω) √ (hij mk − hik mj ) 4f pt f ω Lω{pq + L3 △(sp − qω)} √ (C.ik mj − C.ij mk ). + 2pt f ω

Eij kk − Eik kj = Eij |k − Eik |j +

(5.19)

From (3.17), we get Eij |k − Eik |j = S(hij |k − hik |j ) + T {mi |k mj + mj |k mi −mi |j mk − mk |j mi } + (∂˙k S)hij

−(∂˙j S)hik + (∂˙k T )mi mj − (∂˙j T )mi mk .

Now, (∂˙k S) = −

q (β − ρL2 )s q(f ω2 + f2 ω) √ lk − √ + mk 4L2 (f ω)3/2 2L3 f ω 2L f ω

(5.20)

(5.21)

and (∂˙k T ) = −

sp − qω √ lk + term containing mk , 2p f ω

where we have used the equations (3.7) and (3.8).

From equation (5.9)–(5.11), (5.20) and (5.21), we get Eij |k − Eik |j =

L(sp − qω) √ (C.ij mk − C.ik mj ) 2p f ω q(sp − 2qω) − 2 (hij mk − hik mj ).

4L p(f ω)3/2

(5.22)

Substituting the value of Eij |k − Eik |j from (5.22) in (5.19), then the value of Eij kk − Eik kj thus obtained in (5.17), and then using (3.16) in the right-hand side of (5.17), we find that the equation (5.17) holds identically.

This completes the proof of Codazzi equations of (Mxn , gx ). The Ricci Kühne equations of (Mxn , g x ) consist of the following four equations (a) H (P,Q)i kj − H (P,Q)j ki +

X R

ǫR {H (R,P )i H (R,Q)j

−H (R,P )j H (R,Q)i } + ǫr+1 {H (r+1,P )i H (r+1,Q)j −H (r+1,P )j H (r+1,Q)i } + ǫr+2 {H (r+2,P )i H (r+2,Q)j −H (r+2,P )j H (r+2,Q)i } + ghk {H (P )hi H (Q)kj −H (P )hj H (Q)ki } = 0,

P, Q = 1, 2, · · · , r

(5.23)

β−Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces

(b)

H (P,r+1)i kj − H (P,r+1)j ki +

X R

57

ǫR {H (R,P )i H (R,r+1)j − H (R,P )j H (R,r+1)i }

+ ǫr+2 {H (r+2,P )i H (r+2,r+1)j − H (r+2,P )j H (r+2,r+1)i }

(c)

+ ghk {H (P )hi H (r+1)kj − H (P )hj H (r+1)ki } = 0, P = 1, 2, · · · , r X H (P,r+2)i kj − H (P,r+2)j ki + ǫR {H (R,P )i H (R,r+2)j − H (R,P )j H (R,r+2)i } R

+ ǫr+1 {H (r+1,P )i H (r+1,r+2)j − H (r+1,P )j H (r+1,r+2)i }

+ ghk {H (P )hi H (r+2)kj − H (P )hj H (r+2)ki } = 0, P = 1, 2, · · · , r (d) H (r+1,r+2)i kj − H (r+1,r+2)j ki +

X R

ǫR {H (R,r+1)i H (R,r+2)j − H (R,r+1)j

×H (R,r+2)i } + ghk {H (r+1)hi H (r+2)kj − H (r+1)hj H (r+2)ki } = 0.

Verification of (5.23)(a) In view of (5.1) and (5.2), equation (5.23)a is equivalent to H(P,Q)i kj − H(P,Q)j ki +

X R

ǫR {H(R,P )i H(R,Q)j − H(R,P )j H(R,Q)i }

L3 ω {H(P ).i H(Q).j − H(P ).j H(Q).i } + g hk {H(P )hi H(Q)kj t fp −H(P )hj H(Q)ki } = 0.

P, Q = 1, 2, . . . , r.

L +

(5.24)

Since H(P )ij li = 0 = H(P )ji li , from (3.5), we get g hk {H(P )hi H(Q)kj − H(P )hj H(Q)ki } −H(P )hj H(Q)ki } −

fp = g hk {H(P )hi H(Q)kj L

L3 ω {H(P ).i H(Q).j − H(P ).j H(Q).i }. t

Also, we have H(P,Q)i kj − H(P,Q)j ki = H(P,Q)i |j − H(P,Q)j |i . Hence equation (5.24) is satisfied identically by virtue of (4.3).

Verification of (5.23)(b) In view of (5.1) and (5.2), equation (5.23)b is equivalent to ! ! √ √

L Lω L Lω

√ H(P ).i − √ H(P ).j j i t t √ L Lω X + √ ǫR {H(R,P )i H(R).j − H(R,P )j H(R).i } t R r fp hk + g {H(P )hi dkj − H(P )hj dki } = 0. P, Q = 1, 2, · · · , r.

L

(5.25)

Since bh |j = g hk C.jk , H(P )hi li = 0, we have H(P ).i kj − H(P ).j ki = H(P ).i |j − H(P ).j |i = {H(P )hi |j − H(P )hj |i }bh

−g hk {H(P )hi C.kj − H(P )hj C.ki }

(5.26)

58

O.P.Pandey and H.S.Shukla

√ !

L Lω

√ = ∂˙j j t

√ !

L Lω √ t √ √ 4 L ω Lω L Lω C..j + {pω2 + 3Lω 2 (β − ρL2 )} mj = t3/2 2ωt3/2

(5.27)

and s fp L hk = g × g {H(P )hi dkj − H(P )hj dki } L fp √ L3 ω L {H(P )hi dkj − H(P )hj dki } − √ {H(P ).i d.j − H(P ).j d.i }. t fp hk

r

(5.28)

After using (3.16) and (5.12) the equation (5.28) may be written as √ fp L Lω hk g {H(P )hi dkj − H(P )hj dki } = √ g × L t √ L4 ω Lω {H(P )hi C.kj − H(P )hj C.ki } − {H(P ).i C..j − H(P ).j C..i } t3/2 √ L Lω − [pω2 + 3Lω 2 (β − ρL2 )]{H(P ).i mj − H(P ).j mi }.

2ωt3/2 hk

r

(5.29)

From (4.2), (5.26)–(5.29) it follows that equation (5.25) holds identically.

Verification of (5.23)(c) In view of (5.1) and (5.2), equation (5.23)c is equivalent to √ L Lω(2qω − sp) {H(P ).i mj − H(P ).j mi } √ 2f ωt p r fp hk +g {H(P )hi Ekj − H(P )hj Eki } = 0, L is

(5.30)

Since Ekj lk = 0 = Ejk lk , from (3.5), we find that the value of ghk {H(P )hi Ekj − H(P )hj Eki } s

√ L hk L3 ω L .g {H(P )hi Ekj − H(P )hj Eki } − √ {H(P ).i E.j − H(P ).j E.i }, fp t fp

which, in view of (3.17) and (5.18), is equal to √ L Lω(2qω − sp) − {H(P ).i mj − H(P ).j mi }. √ 2f ωt p Hence equation (5.30) is satisfied identically.

Verification of (5.23)(d) In view of (5.1) and (5.2), equation (5.23)d is equivalent to (N mi )kj − (N mj )ki + ghk (dhi Ekj − dhj Eki ) = 0, where N =

sp−2qω √ .

2f ω pt

(5.31)

β−Change of Finsler Metric by h-Vector and Imbedding Classes of Their Tangent Spaces

59

Since dhi lh = 0, Ekj lk = 0, from (3.5), we find that the value of ghk {dhi Ekj − dhj Eki } is L hk L4 ω g {dhi Ekj − dhj Eki } − {d.i E.j − d.j E.i }, fp f pt which, in view of (3.16), (3.17), (5.12) and (5.18), is equal to −

L3 (2qω − sp) {C..i mj − C..j mi }. √ 2f p.t3/2

Also, (N mi )kj − (N mj )ki = N (mi kj − mj ki ) + (∂˙j N )mi − (∂˙i N )mj .

Since mi kj − mj ki = mi |j − mj |i = L−1 (lj mi − li mj ) and 2qω − sp L3 (sp − 2qω) √ lj + ∂˙j N = − C..j , √ 2Lf ω pt 2f p.t3/2 we have (N mi )kj − (N mj )ki =

L3 (sp − 2qω) (C..j mi − C..i mj ). √ 2f p.t3/2

(5.32)

Hence equation (5.31) is satisfied identically. Therefore Ricci Kühne equations of (Mxn , gx ) given in (5.23) are satisfied.

Hence the Theorem A given in introduction is satisfied for the β−change (1.3) of Finsler metric given by h−vector.

2 References [1] Eisenhart L. P., Riemannian Geometry, Princeton, 1926.

[2] Matsumoto M., On transformations of locally Minkowskian space, Tensor N. S., 22 (1971), 103-111.

[3] Prasad B. N., Shukla H. S. and Pandey O. P., The exponential change of Finsler metric and relation between imbedding class numbers of their tangent Riemannian spaces, Romanian Journal of Mathematics and Computer Science, 3 (1), (2013), 96-108.

[4] Prasad B. N., Shukla H. S. and Singh D. D., On a transformation of the Finsler metric, Math. Vesnik, 42 (1990), 45-53.

[5] Prasad B. N. and Kumari Bindu, The β−change of Finsler metric and imbedding classes of their tangent spaces, Tensor N. S., 74 (1), (2013), 48-59.

[6] Singh U. P., Prasad B. N. and Kumari Bindu, On a Kropina change of Finsler metric, Tensor N. S., 64 (2003), 181-188.

[7] Shibata C., On invariant tensors of β−changes of Finsler metrics, J. Math. Kyoto University, 24 (1984), 163-188.