Some Properties of Conformal Beta-Change

International J.Math. Combin. Vol.2(2018), 13-23

Some Properties of Conformal β -Change H.S.Shukla and Neelam Mishra (Department of Mathematics and Statistics, D. D. U. Gorakhpur University, Gorakhpur (U.P.)-273009,India) E-mail: profhsshuklagkp@rediffmail.com, pneelammishra@gmail.com

Abstract: We have considered the conformal β-change of the Finsler metric given by L(x, y) → L̄(x, y) = eσ(x) f (L(x, y), β(x, y)), where σ(x) is a function of x, β(x, y) = bi (x)y i is a 1-form on the underlying manifold M n ,and f (L(x, y), β(x, y)) is a homogeneous function of degree one in L and β.We have studied quasi-C-reducibility, C-reducibility and semi-C-reducibility of the Finsler space with this metric. We have also calculated V-curvature tensor and T-tensor of the space with this changed metric in terms of v-curvature tensor and T-tensor respectively of the space with the original metric.

Key Words:

Conformal change, β-change, Finsler space, quasi-C-reducibility, C-

reducibility, semi-C-reducibility, V-curvature tensor, T-tensor.

AMS(2010): 53B40, 53C60. §1. Introduction Let F n = (M n , L) be an n-dimensional Finsler space on the differentialble manifold M n equipped with the fundamental function L(x,y).B.N.Prasad and Bindu Kumari and C. Shibata [1,2] have studied the general case of β-change,that is, L∗ (x, y) = f (L, β),where f is positively homogeneous function of degree one in L and β, and β given by β(x, y) = bi (x)y i is a one- form on M n . The β-change of special Finsler spaces has been studied by H.S.Shukla, O.P.Pandey and Khageshwar Mandal [7].

The conformal theory of Finsler space was initiated by M.S. Knebelman [12] in 1929 and has been investigated in detail by many authors (Hashiguchi [8] ,Izumi[4,5] and Kitayama [9]).

The conformal change is defined as L∗ (x, y) = eσ(x) L(x, y), where σ(x) is a function of position only and known as conformal factor. In 2008, Abed [15,16] introduced the change L̄(x, y) = eσ(x) L(x, y) + β(x, y), which he called a β-conformal change, and in 2009 and 2010,Nabil L.Youssef, S.H.Abed and S.G. Elgendi [13,14] introduced the transformation L̄(x, y) = f (eσ L, β), which is β-change of conformally changed Finsler metric L. They have not only established the relationships between some important tensors of (M n , L) and the corresponding tensors of (M n , L̄), but have also studied several properties of this change.

1 Received

January 22, 2018, Accepted May 12, 2018.

14

H.S.Shukla and Neelam Mishra

We have changed the order of combination of the above two changes in our paper [6], where we have applied β-change first and conformal change afterwards, i.e., L̄(x, y) = eσ(x) f (L(x, y), β(x, y)),

(1.1)

where σ(x) is a function of x, β(x, y) = bi (x)y i is a 1-form. We have called this change as conformal β-change of Finsler metric. In this paper we have investigated the condition under which a conformal β-change of Finsler metric leads a Douglas space into a Douglas space.We have also found the necessary and sufficient conditions for this change to be a projective change.

In the present paper,we investigate some properties of conformal β-change. The Finsler space equipped with the metric L̄ given by (1.1) will be denoted by F¯n .Throughout the paper the quantities corresponding to F¯n will be denoted by putting bar on the top of them.We shall denote the partial derivatives with respect to xi and y i by ∂i and ∂˙i respectively. The Fundamental quantities of F n are given by 2

L gij = ∂˙i ∂˙j = hij + li lj , li = ∂˙i L. 2 Homogeneity of f gives Lf1 + βf2 = f,

(1.2)

where subscripts 1 and 2 denote the partial derivatives with respect to L and β respectively.

Differentiating above equations with respect to L and β respectively, we get Lf12 + βf22 = 0 and Lf11 + βf21 = 0.

(1.3)

f11 /β 2 = (−f12 )/Lβ = f22 /L2 ,

(1.4)

f11 = β 2 ω, f12 = −Lβω, f22 = L2 ω,

(1.5)

Hence we have

which gives

where Weierstrass function ω is positively homogeneous of degree -3 in L and β. Therefore Lω1 + βω2 + 3ω = 0,

(1.6)

where ω1 and ω2 are positively homogeneous of degree -4 in L and β. Throughout the paper we frequently use the above equations without quoting them. Also we have assumed that f is not linear function of L and β so that ω 6= 0.

The concept of concurrent vector field has been given by Matsumoto and K. Eguchi [11] and S. Tachibana [17], which is defind as follows: The vector field bi is said to be a concurrent vector field if bi|j = −gij

bi |j = 0,

(1.7)

where small and long solidus denote the h- and v-covariant derivatives respectively. It has been

Some Properties of Conformal β -Change

15

proved by Matsumoto that bi and its contravariant components bi are functions of coordinates alone. Therefore from the second equation of (1.7),we have Cijk bi = 0.

The aim of this paper is to study some special Finsler spaces arising from conformal β-change of Finsler metric,viz., quasi-C-reducible, C-reducible and semi-C-reducible Finsler spaces. Further, we shall obtain v-curvature tensor and T-tensor of this space and connect them with v-curvature tensor and T-tensor respectively of the original space.

§2. Metric Tensor and Angular Metric Tensor of F̄ n Differentiating equation (1.1) with respect to y i we have l̄i = eσ (f1 li + f2 bi ).

(2.1)

Differentiating (2.1) with respect to y j , we get h̄ij = e2σ

where mi = bi −

f f1 hij + f L2 ωmi mj , L

(2.2)

β Li .

L

From (2.1) and (2.2) we get the following relation between metric tensors of F n and F̄ n : ḡij = e2σ

f f1 pβ gij − li lj + (f L2 ω + f22 )bi bj + p(bi lj + bj li ) , L L

(2.3)

where p = f1 f2 − f βLω.

The contravariant components ḡ ij of the metric tensor of F̄ n , obtainable from ḡ ij ḡjk = δki , are as follows:

L ij pL3 fβ L4 ω i j pL2 i j ij −2σ i j j i ḡ = e g + 3 − ∆f2 l l − b b − 2 (l b + l b ) , (2.4) f f1 f f1 t L 2 f f1 t f f1 t where li = g ij lj , b2 = bi bi , bi = g ij bj , g ij is the reciprocal tensor of gij of F n , and t = f1 + L3 ω∆, ∆ = b2 −

β2 .

L2

f li + f2 mi , (b) ∂˙i f1 = −eσ βLωmi , L σ 2 (c)∂˙i f2 = e L ωmi , (d) ∂˙i p = −βqLmi , 3ω (e) ∂˙i ω = − li + ω2 mi , (f )∂˙i b2 = −2C..i , L 2β (g) ∂˙i ∆ = −2C..i − 2 mi , L

(a) ∂˙i f = eσ

(2.5)

(2.6)

16

H.S.Shukla and Neelam Mishra

3q (a) ∂˙i q = − li , (b) ∂˙i t = −2L3 ωC..i + [L3 ∆ω2 − 3βLω]mi , L 3q (c) ∂˙i q = − li + (4f2 ω2 + 3ω 2 L2 + f ω22 )mi .

L

(2.7)

§3. Cartan’s C-Tensor and C-Vectors of F̄ n Cartan’s covariant C-tensor Cijk of F n is defined by C̄ijk =

1˙ ˙ ˙ 2 ∂i ∂j ∂k L = ∂˙k gij 4

and Cartan’s C-vectors are defined as follows: i jk Ci = Cijk g jk , C i = Cjk g .

(3.1)

We shall write C 2 = C i Ci . Under the conformal β-chang (1.1) we get the following relation between Cartan’s C-tensors of F n and F̄ n :

p qL2 2σ f f1 C̄ijk = e Cijk + (hij mk + hjk mi + hki mj ) + mi mj mk .

(3.2) L 2L 2 We have (a) mi li = 0, β2 = ∆ = b i mi , L2 (c) gij mi = hij mi = mj .

(b) mi bi = b2 −

(3.3)

From (2.1), (2.3), (2.4) and (3.2), we get h C̄ij

=

qL3 p (hij mh + hhj mi + hhi mj ) + mj mk mh 2f f1 2f f1 L pL∆ 2pL + qL4 ∆ − C.jk nh − 2 hjk nh − mj mk n h , ft 2f f1 t 2f 2 f1 t h Cij +

(3.4)

where nh = f L2 ωbh + plh and hij = g il hlj , C.ij = Crij br , C..i = Crji br bj and so on.

Proposition 3.1 Let F̄ n = (M n , L̄) be an n-dimensional Finsler space obtained from the conformal β-change of the Finsler space F n = (M n , L), then the normalized supporting element l̄i , angular metric tensor h̄ij , fundamental metric tensor ḡij and (h)hv-torsion tensor C̄ijk of F̄ n are given by (2.1), (2.2), (2.3) and (3.2), respectively.

F

n

From (2.4),(3.1),(3.2) and (3.4) we get the following relations between the C-vectors of of and F̄ n and their magnitudes C̄i = Ci − L3 ωCi.. + µmi ,

(3.5)

17

Some Properties of Conformal β -Change

where µ=

p(n + 1) 3pL3 ω∆ qL3 ∆(1 − L3 ω∆) − + ; 2f f1 2f f1 2f f1 C̄ i =

where

e−2σ L i C + M i, f f1

µe−2σ L i L4 ω i m − C − Ci − e2σ L3 ωCi.. + µ∆ M = f f1 f f1 .. i

and

CÌ„ 2 =

(3.6)

L3 ω i L b + yi f f1 ft

e−2σ 2 C + λ, p

(3.7)

where

λ =

e−2σ L 2µe−2σ L 3 − L ω∆ µ2 ∆ + C. f f1 f f1

− (1 + 2µ∆) L3 ω + 1 − 3µ + e2σ L2 ωf f1 C. L3 ωC...

+L3 ωC..r e4σ Lωf 2 f12 Ci.. − µ∆ L3 ωbr − e2σ L2 ωf f1 C..r − 2C r .

§4. Special Cases of F̄ n

In this section, following Matsumoto [10], we shall investigate special cases of F̄ n which is conformally β-changed Finsler space obtained from F n .

Definition 4.1 A Finsler space (M n , L) with dimension n ≥ 3 is said to be quasi-C-reducible if the Cartan tensor Cijk satisfies Cijk = Qij Ck + Qjk Ci + Qki Cj ,

(4.1)

where Qij is a symmetric indicatory tensor.

The equation (3.2) can be put as C̄ijk = e2σ

f f1 1 Cijk + π(ijk) L 6

3p hij + qL2 mi mj mk , L

where π(ijk) represents cyclic permutation and sum over the indices i, j and k.

Putting the value of mk from equation (3.5) in the above equation, we get CÌ„ijk = e

2σ

f f1 1 Cijk + π(ijk) L 6µ

3p hij + qL2 mi mj )(C̄k − Ck + L3 ωCk..

L

.

18

H.S.Shukla and Neelam Mishra

Rearranging this equation, we get CÌ„ijk

=

f f1 1 3p Cijk + π(ijk) hij + qL2 mi mj C̄k L 6µ L

1 3p + π(ijk) hij + qL2 mi mj L3 ωCk.. − Ck .

6µ L e2σ

Further rearrangment of this equations gives C̄ijk = π(ijk) (H̄ij C̄k ) + Uijk , where H̄ij =

3p e2σ 6µ {( L hij

Uijk = e

2σ

(4.2)

+ qL2 mi mj ), and

f f1 1 Cijk + π(ijk) L 6µ

3p hij + qL2 mi mj L

3

L ωCk.. − Ck

(4.3)

Since HÌ„ij is a symmetric and indicatory tensor,therefore from equation (4.2) we have the following theorem.

Theorem 4.1 Conformally β-changed Finsler space F̄ n is quasi-C-reducible iff the tensor Uijk of equation (4.3) vanishes identically.

We obtain a generalized form of Matsumoto’s result [10] as a corollary of the above theorem.

Corollary 4.1 If F n is Reimannian space, then the conformally β-changed Finsler space F̄ n is always a quasi-C-reducible Finsler space.

Definition 4.2 A Finsler space (M n , L) of dimension n ≥ 3 is called C-reducible if the Cartan tensor Cijk is written in the form Cijk =

1 (hij Ck + hki Cj + hjk Ci ). n+1

(4.4)

1 Define the tensor Gijk = Cijk − (n+1) (hij Ck + hki Cj + hjk Ci ). It is clear that Gijk is symmetric and indicatory. Moreover, Gijk vanishes iff F n is C-reducible.

Proposition 4.1 Under the conformal β-change(1.1), the tensor Ḡijk associated with the space F̄ n has the form f f1 Ḡijk = e2σ Gijk + Vijk (4.5) L where Vijk

=

1 π(ijk) {(e2σ (n + 1)(α1 hij + α2 mi mj )mk + e2σ ωL2 mi mj Ck (n + 1) +e2σ L2 ω(f f1 hij + L3 ωmi mj )Ck.. }, α1 =

e2σ p µf f1 e2σ − , 2L L(n + 1)

α2 =

(4.6) e2σ qL2 µe2σ ωL2 − .

6 (n + 1)

Some Properties of Conformal β -Change

19

From (4.5) we have the following theorem.

Theorem 4.2 Conformally β-changed Finsler space F̄ n is C-reducible iff F n is C-reducible and the tensor Vijk given by (4.6) vanishes identically.

Definition 4.3 A Finsler space (M n , L) of dimension n ≥ 3 is called semi-C-reducible if the Cartan tensor Cijk is expressible in the form: Cijk =

r s (hij Ck + hki Cj + hjk Ci ) + 2 Ci Cj Ck , n+1 C

(4.7)

where r and s are scalar functions such that r + s = 1.

Using equations (2.2), (3.5) and (3.7) in equation (3.2), we have C̄ijk = e2σ

f f1 p ∆L(f1 q − 3pω) Cijk + (h̄ij C̄k + h̄ki C̄j + h̄jk C̄i ) + C̄ C̄ C̄ i j k .

L 2µf f1 2f f1 µtC̄ 2

If we put p(n + 1) ′ ∆L(f1 q − 3pω) ,s = , 2µf f1 2f f1 µt

r′ =

we find that r′ + s′ = 1 and

f f1 r′ s′ C̄ijk = e2σ Cijk + (h̄ij C̄k + h̄ki C̄j + h̄jk C̄i ) + 2 C̄i C̄j C̄k .

L n+1 C̄

(4.8)

From equation (4.8) we infer that F̄ n is semi-C- reducible iff Cijk = 0, i.e. iff F n is a Reimannian space. Thus we have the following theorem.

Theorem 4.3 Conformally β-changed Finsler space F̄ n is semi-C-reducible iff F n is a Riemannian space.

§5. v-Curvature Tensor of F̄ n The v-curvature tensor [10] of Finsler space with fundamental function L is given by r r Shijk = Cijr Chk − Cikr Chj

Therefore the v-curvature tensor of conformally β-changed Finsler space F̄ n will be given by r r S̄hijk = C̄ijr C̄hk − C̄ikr C̄hj .

From equations (3.2)and(3.4), we have r C̄ijr C̄hk

f f1 p r Cijr Chk + (Cijk mh + Cijh mk + Cihk mj L 2L pf1 f f1 L 2 ω +Chjk mi ) + (C.ij hhk + Chk hij ) − C.ij C.hk 2Lt t

= e2σ

(5.1)

20

H.S.Shukla and Neelam Mishra

p2 ∆ L2 (qf1 − 2pω) hhk hij + (C.ij mk mh + C.hk mi mj ) 4f Lt 2t p(p + L3 q∆) p2 + (hij mh mk + hhk mi mj ) + (hij mh mk 4Lf t 4Lf f1 +hhk mi mj + hhj mi mk + hhi mj mk + hjk mi mh + hik mh mj )

L2 (2pqt + (qf1 − 2pω)(2p + L3 q∆)) mi mj mh mk . + 4f f1 t +

(5.2)

We get the following relation between v-curvature tensors of (M n , L) and (M n , L̄): S̄hijk = e2σ

f f1 Shijk + dhj dik − dhk dij + Ehk Eij − Ehj Eik , L

(5.3)

where

P =L

s 1/2 t

, Q=

dij = P C.ij − Qhij + Rmi mj ,

(5.4)

Eij = Shij + T mi mj ,

(5.5)

pg L (2ωp − f1 q) p L (qf1 − ωp) √ ,R = √ √ , T = √ ,S = .

2L2 f ω 2f1 f ω 2L2 st 2 st

Proposition 5.1 The relation between v-curvature tensors of F n and F̄ n is given by (5.3).

When bi in β is a concurrent vector field,then C.ij = 0. Therefore the value of v-curvature tensor of F̄ n as given by (5.3) is reduced to the extent that dij = Rmi mj − Qhij . §6. The T-Tensor Thijk The T-tensor of F n is defined in [3] by Thijk = LChij |k +Chij lk + Chik lj + Chjk li + Cijk lh ,

(6.1)

r r r Chij |k = ∂˙k Chij − Crij Chk − Chrj Cik − Chir Cjk .

(6.2)

where In this section we compute the T-tensor of F̄ n , which is given by T̄hijk = L̄C̄hij¯|k + C̄hij l̄k + C̄hik ¯lj + C̄hjk l̄i + C̄ijk l̄h ,

(6.3)

r r r C̄hij¯|k = ∂˙k C̄hij − C̄rij C̄hk − C̄hrj C̄ik − C̄hir C̄jk .

(6.4)

where The derivatives of mi and hij with respect to y k are given by β 1 ∂˙k mi = − 2 hik − (li mk ), L L

1 ∂˙k hij = 2Cijk − (li hjk + lj hki ) L

(6.5)

Some Properties of Conformal β -Change

21

From equations (3.2)and (6.5), we have ∂˙k C̄hij

f f1 p ∂k Chij + (Cijk mh + Cijh mk + Cihk mj + Chjk mi ) L L pβ p − 3 (hij hhk + hhj hik + hih hjk ) + (hjk lh mi + hhk lj mi 2L 2L2 +hhk li mj + hik lh mj + hjk li mh + hjk lh mi + hij lh mk + hhj li mk βq (hij mh mk +hik lj mk + hij lk mh + hjh lk mi + hhi lk mj ) − 2 +hhk mi mj + hhj mi mk + hhi mj mk + hjk mi mh + hik mh mj ) qL − (li mj mh mk + lj mi mh mk + lh mi mj mk + hk mi mj mh ) 2

L2 + (4f2 ω2 + 3L2 ω 2 + f ω22 )mh mi mj mk .

2

= e2σ

(6.6)

Using equations (6.5) and (5.2) in equation(6.4), we get C̄hij¯|k

e2σ p f f1 Chij |k − (Cijk mh + Cijh mk + Cihk mj + Chjk mi ) L 2L

2f βt L2 p∆ βq 2σ 2σ −pe + (hij hhk + hhj hik + hih hjk ) − e 3 3 4f L t 4f L t 2

p2 f1 + pqf1 L3 ∆ + 3p2 + (hij mh mk + hhk mi mj + hhj mi mk + hhi mj mk 4Lf f1t e2σ p +hjk mi mh + hik mh mj ) − [lh (hjk mi + hij mk 2L2 +hik mj ) + lj (hhk mi + hik mkh + hih mk ) + li (hhk mj + hjk mh

= e2σ

e2σ qL (li mj mh mk 2 pf1 e2σ +lj mi mh mk + lh mi mj mk + hk mi mj mh ) − (C.ij hhk 2Lt e2σ f f1 L2 ω +C.hj hik + C.hk hij + C.ik hh + C.hi hjk + C.jk hhi ) + (C.ij C.hk t 2σ 2 e L (qf1 − 2pω) +C.hj C.ik + C.hi C.jk ) − (C.ij mk mh 2t +C.hk mi mj + C.hj mi mk + C.ik mj mh 2 2 2 2σ L (4f2 ω2 + 3L ω + f ω22 ) +C.hi mj mk + C.jk mh mi ) + e 2

2 3 3L (2pqt + (qf1 − 2pω)(2p + L q∆) − mi mj mh mk .

(6.7) 4f f1 t +hhj mk ) + lk (hij mh + hjh mi + hhi mj )] −

Using equations (2.1), (3.2) and (6.6) in equation (6.3), we get the following relation

22

H.S.Shukla and Neelam Mishra

between T-tensors of Finsler spaces F n and F̄ n : T̄hijk

f 2 f1 f (f1 f2 + f βLω) Thijk + (Cijk mh + Cijh mk + Cihk mj L2 2L f 2 f1 L 2 ω (C.ij C.hk + C.hj C.ik + C.hi C.jk ) +Chjk mi ) + t pf1 − (C.ij hhk + C.hj hik + C.hk hij + C.ik hh + C.hi hjk + C.jk hhi ) 2Lt f L2 (qf1 − 2pω) − (C.ij mk mh + C.hk mi mj + C.hj mi mk 2t p(2f βt + L2 p∆) (hij hhk +C.ik mj mh + C.hi mj mk + C.jk mh mi ) − 4L3 t 2

p f1 + pqf1 L3 ∆ + 3p2 βqf pf2 + − +hhj hik + hih hjk ) − 4Lf1t 2 L (hij mh mk + hhk mi mj + hhj mi mk + hhi mj mk + hjk mi mh 2 L (4f2 ω2 + 3L2 ω 2 + f ω22 ) +hik mh mj ) + + 2L2 f2 q 2

3L2 (2pqt + (qf1 − 2pω)(2p + L3 q∆) − mi mj mh mk .

4f1 t

= e3σ

(6.8)

Proposition 6.1 The relation between T-tensors of F n and F̄ n is given by (6.7).

If bi is a concurrent vector field in F n , then C.ij = 0. Therefore from(6.8), we have T̄hijk

f 2 f1 p(2f βt + L2 p∆) T − (hij hhk + hhj hik + hih hjk ) hijk L2 4L3 t

2 pf2 p f1 + pqf1 L3 ∆ + 3p2 t βqf + − (hij mh mk + hhk mi mj − 4Lf1 t 2 L +hhj mi mk + hhi mj mk + hjk mi mh + hik mh mj )

L2 (4f2 ω2 + 3L2 ω 2 + f ω22 ) 3L2 (qf1 − 2pω)(2p + L3 q∆) + 2L2 f2 q + + 2 4Lf f1t

2 3L 2pqt − mi mj mh mk .

4Lf f1 t

= e

3σ

(6.9)

If bi is a concurrent vector field in F n , with vanishing T-tensor then T-tensor of F n is given by T̄hijk

p(2f βt + L2 p∆) (hij hhk + hhj hik + hih hjk ) 4L3 t 2

p f1 + pqf1 L3 ∆ + 3p2 t βqf pf2 − + − (hij mh mk 4Lf1 t 2 L +hhk mi mj + hhj mi mk + hhi mj mk + hjk mi mh + hik mh mj ) 2 L (4f2 ω2 + 3L2 ω 2 + f ω22 ) 3L2 2pqt + − 2 4Lf f1t

2 3 3L (qf1 − 2pω)(2p + L q∆) 2 + + 2L f2 q mi mj mh mk .

4Lf f1t

= e

3σ

−

(6.10)

Some Properties of Conformal β -Change

23

Acknowledgement The work contained in this research paper is part of Major Research Project “Certain Investigations in Finsler Geometry” financed by the U.G.C., New Delhi.

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

[2] C. Shibata, On invariant tensors of β-change of Finsler metric, J. Math.Kyoto Univ., 24(1984),163-188.

[3] F.Ikeda, On the tensor Tijkl of Finsler spaces, Tensor N. S., 33(1979),203-209.

[4] H.Izumi, Conformal transformations of Finsler spaces I, Tensor N.S., 31(1977),33-41.

[5] H.Izumi, Conformal transformations of Finsler spaces II. An h-conformally flat Finsler space, Tensor N.S., 33(1980),337-369.

[6] H.S.Shukla and Neelam Mishra, Conformal β-changes of Finsler metric, J.Int.Acad.Phys.Sci., 21(1)(2017), 19-30.

[7] H.S.Shukla, O.P.pandey and Khageshwar Mandal, The β-change of Special Finsler Spaces, International J.Math.Combin., 1(2017), 78-87.

[8] M.Hashiguchi, On conformal transformation of Finsler metric, J.Math.Kyoto Univ.,16(1976),2550.

[9] M.Kitayama, Geometry of Transformations of Finsler metrics, Hokkaido University of Education,Kushiro Campus, Japan, 2000.

[10] M.Mastsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Saikawa,Otsu, 520 Japan, 1986.

[11] M. Matsumoto and K. Eguchi, Finsler space admitting a concurrent vector field, Tensor N.S., 28(1974), 239-249.

[12] M.S.Knebelman, Conformal geometry of generalized metric spaces, Proc.Nat.Acad. Sci., USA.,15(1929),376-379.

[13] N.L.Youssef, S.H.Abed and S.G.Elgendi, Generalized β-conformal change of Finsler metrics, Int.J.Geom.Meth.Mod.Phys.,7,4(2010).ArXiv No.:math.DG/0906.5369.

[14] N.L.Youssef,S.H.Abed and S.G.Elgendi, Generalized β-conformal change and special Finsler spaces, Int.J.Geom.Meth.Mod.Phys., 8,6(2010). ArXiv No.:math.DG/1004.5478v3.

[15] S.H.Abed, Conformal β-changes in Finsler spaces, Proc.Math.Phys.Soc.Egypt, 86(2008),7989.ArXiv No.:math.DG/0602404.

[16] S.H.Abed, Cartan connections associated with a β-conformal change in Finsler geometry, Tensor N.S., 70(2008),146-158. ArXiv No.:math.DG/0701491.

[17] S.Tachibana, On Finsler spaces which admit a concurrent vector field, Tensor N. S., 1 (1950),1-5.