On the conformal change of five-dimensional Finsler spaces

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1 On the conformal change of five-dimensional Finsler spaces Gauree Shanker Abstract. The purpose of the present paper is to deal with the theory of conformal change in five-dimensional Finsler space. We have obtained the conditions under which the h- and v- connection vectors are conformally invariant in five-dimensional Finsler space. M.S.C. 2010: 53B40, 53C60. Key words: Finsler space; Conformal change; h and v-connection vectors; main scalars. 1 Introduction 9 The conformal change and conformal transformation of n-dimensional Finsler spaces 10 have been studied in [6] and [1]. The conformal theory of two, three & four-dimensional 11 Finsler spces have been discussed in [2, 8, 10] respectively. As far as author knows 12 there is no paper concerned with the conformal theory of five-dimensional Finsler 13 space. Recently, the present author has found that in a five-dimensional Finsler space there are seventeen main scalars H, I, J, K, H, I, J, K, H, I, J, K, M, M, M 14, N, N 15 [11] in which the sum of H, I, K and M is LC, which is called unified main 16 scalar. It has been also shown by the present author that in a five-dimensional Finsler space there exist six v-connection vectors u i, v i, w i, u 17 i, v i, w i and six h-connection vectors h i, J i, k i, h i, J i 18, k i [11]. The theory of five-dimensional Finsler space with con- stant unified main scalars has been discussed in [12]. The orthonormal frame field ( l i, m i, n i, p i, q i), called the Miron frame plays an important role in five-dimensional Finsler space. Here we discuss the theory of conformal change in five-dimensional Finsler space. 2 Scalar components in Miron frame Let F 5 be a five-dimensional Finsler space with fundamental function L(x, y). The metric tensor g ij and Cartan C tensor C ijk of F 5 are defined by g ij = 1 2 i j L2, C ijk = 1 2 k g ij = 1 4 i j k L2. Differential Geometry - Dynamical Systems, Vol.15, 2013, pp c Balkan Society of Geometers, Geometry Balkan Press 2013.

2 80 Gauree Shanker Throughout the paper, the symbols i { } = frame e i (α) y and i i = x i have been used. The, α = 1, 2, 3, 4, 5 is called the Miron frame of F 5, where e i (1) = li = yi L 29 is called the normalized supporting element, e i (2) = mi = Ci C is called the normalized 30 torsion vector, e i (3) = ni, e i (4) = pi, e i (5) = qi are constructed from g ij e i (α) ej (β) = 31 δ αβ.here, C is the length of torsion vector C i = C ijkg jk.the Greek letters α, β, γ, δ 32 vary from 1 to 5 throughout the paper. Summation convention is applied for both 33 the Greek and Latin indices. In the Miron frame an arbitrary tensor can be expressed 34 by scalar components along the unit vectors l i, m i, n i, p i, q i. For instance; let T = Tj i 35 be a tensor field of (1, 1) type, then the scalar components T αβ of T are defined by T αβ = Tj ie (α)ie j (β) and the components T j i of the tensor T are expressed as T j i = 37 T αβe i (α) e (β)j.from the equation g ij e i (α) ej (β) = δ αβ, we have (2.1) g ij = l i l j + m i m j + n i n j + p i p j + q i q j Next, the C-tensor C ijk = 1 2 k g ij satisfies C ijk l i = 0 and is symmetric in i, j, k. Therefore, if C αβγ are scalar components of LC ijk i. e., if (2.2) LC ijk = C αβγ e (α)i e (β)j e (γ)k, then, we have (2.3) LC ijk = C 222 m i m j m k + C 223 (m i m j n k ) + C 233 (m i n j n k ) + C 333 (n i n j n k ) + C 224 (m i m j p k ) + C 444 (p i p j p k ) + C 244 (m i p j p k ) + C 225 (m i m j q k ) + C 255 (m i q j q k ) + C 555 (q i q j q k ) + C 334 (n i n j p k ) + C 344 (n i p j p k ) + C 335 (n i n j q k ) + C 355 (n i q j q k ) + C 445 (p i p j q k ) + C 455 (p i q j q k ) + C 234 {m i (n j p k + n k p j )} + C 235 {m i (n j q k + n k q j )} + C 245 {m i (p j q k + p k q j )} + C 345 {n i (p j q k + p k q j )}, where 38 {...} denote the cyclic interchange of i, j, k and summation. For instance; {A ib j C k } = A i B j C k + A j B k C i + A k B i C j. 39

3 On the conformal change of five-dimensional Finsler spaces 81 Contracting (2.2) with g jk, we get LCm i = C αββ e (α)i. Thus, if we put (2.4) C 222 = H, C 233 = I, C 244 = K, C 333 = J, C 344 = J, C 444 = H, C 334 = I, C 234 = K, C 255 = M, C 355 = J, C 455 = M, C 555 = H, C 335 = I, C 445 = K, C 235 = N, C 245 = N, C 345 = M, C 224 = (H + I + M ), C 225 = ( H + I + K ). then, we have (2.5) H + I + K + M = LC, C 223 = (J + J + J ), Hence, we have the following: Theorem 2.1. In a five-dimensional Finsler space there are seventeen main scalars H, I, J, K, H, I, J, K, H, I, J, K, M, M, M, N, N in which the sum of H, I, K and M is LC which is called unified main scalar. Using (2.4) and (2.5), the equation (2.3) can be rewritten as [11] (2.6) LC ijk = Hm i m j m k (J + J + J ) (m i m j n k ) + I (m i n j n k ) J (n i n j n k ) (H + I + M ) (m im j p k ) + H (p i p j p k ) +K (m ip j p k ) (H + I + K ) (m im j q k ) +M (m iq j q k ) + H (q i q j q k ) + I (n in j p k ) + J (n ip j p k ) +I (n in j q k ) + J (n iq j q k ) + K (p ip j q k ) +M (p iq j q k )+K {m i (n j p k + n k p j )}+ N {m i (n j q k + n k q j )} +N {m i (p j q k + p k q j )}+M {n i (p j q k + p k q j )}. ( ) The Cartan s connection CΓ = Γ i jk, Gi j, Ci jk will be used in the following section of this paper. The h and v covariant derivatives of the frame field e (α)i are given by [4] (2.7) e (α)i j=h(α)βγ e (β)i e (γ)j, Le (α)i j = V (α)βγ e (β)i e (γ)j, where H (α)βγ and V (α)βγ, γ being fixed, are given by (2.8) H α)βγ = h γ J γ k γ 0 h γ 0 h γ J γ 0 J γ h γ 0 k γ 0 k γ J γ k γ 0, V α)βγ = 0 δ 2γ δ 3γ δ 4γ δ 5γ δ 2γ 0 u γ v γ w γ δ 3γ u γ 0 u γ v γ δ 4γ v γ u γ 0 w γ δ 5γ w γ v γ w γ 0 In (2.8), we have put (2.9) H 2)3γ = H 3)2γ = h γ, H 2)4γ = H 4)2γ = J γ, H 2)5γ = H 5)2γ = k γ,

4 82 Gauree Shanker H 3)4γ = H 4)3γ = h γ, H 3)5γ = H 5)3γ = J γ, H 4)5γ = H 5)4γ = k γ, V 2)3γ = V 3)2γ = u γ, V 2)4γ = V 4)2γ = v γ, V 2)5γ = V 5)2γ = w γ, V 3)4γ = V 4)3γ = u γ, V 3)5γ = V 5)3γ = v γ, V 4)5γ = V 5)4γ = w γ. Hence, we have the following: 54 Theorem 2.2. In a five-dimensional Finsler space there exist six h-connection} vectors h i, J i, k i, h i, J i, k i {e whose scalar components with respect to the frame i 55 (α) are h γ, J γ, k γ, h γ, J γ, k γ, i. e., h i = h γ e (γ)i, J i = J γ e (γ)i, k i = k γ e (γ)i, h i = h γe (γ)i, J i 56 = J γe (γ)i, k i 57 = k γe (γ)i. 58 Theorem 2.3. In a five-dimensional Finsler space there exist six v-connection} vectors u i, v i, w i, u i, v i, w i {e whose scalar components with respect to the frame i 59 (α) are u γ, v γ, w γ, u γ, v γ, w γ i. e., u i = u γ e (γ)i, v i = v γ e (γ)i, w i = w γ e (γ)i, u i = u γe (γ)i, v i 60 = v γe (γ)i, w i 61 = w γe (γ)i. In view of equations (2.8), (2.9) and using the theorems (2.2) and (2.3), the equations (2.7) may be explicitly written as [11] (2.10) l i j = 0, m i j = n i h j + p i J j + q i k j, n i j = m i h j + p i h j + q i J j, p i j = m i J j n i h j + q i k j, q i j = m i k j n i J j p i k j and (2.11) Ll i j = m i m j + n i n j + p i p j + q i q j = g ij l i l j = h ij, Lm i j = l i m j + n i u j + p i v j + q i w j, Ln i j = l i n j m i u j + p i u j + q i v j, Lp i j = l i p j m i v j n i u j + q i w j, Lq i j = l i q j m i w j n i v j p i w j. 62 Since m i, n i, p i, q i are homogeneous functions of degree zero in y i, we have Lm i j l j = Ln i j l j = Lp i j l j = Lq i j l j = 0 which in view of equation (2.11) and theorem (2.3) gives (2.12) u 1 = v 1 = w 1 = u 1 = v 1 = w 1 = Thus we have the following: Theorem 2.4. In a five-dimensional Finsler space, the first scalar components of v-connection vectors u i, v i, w i, u i, v i, w i vanish identically. The equations (2.11) and (2.6) lead to the following expressions for the partial derivatives with respect to y j : (2.13) L jl i = h ij = m i m j + n i n j + p i p j + q i q j, 66 L j m i = l i m j + n i u j + p i v j + q i w j + Hm i m j + In i n j + Kp i p j + Mq i q j

5 On the conformal change of five-dimensional Finsler spaces (J + J + J )(m i n j + m j n i ) (H + I + M )(m i p j + m j p i ) (H + I + K )(m i q j + m j q i ) + K (n i p j + n j p i ) + N(n i q j + n j q i ) +N (p i q j + p j q i ), L j n i = l i n j m i u j + p i u j + q iv j (J + J + J )m i m j + Jn i n j + J p i p j +J q i q j + I(m i n j + m j n i ) + K (m i p j + m j p i ) + N(m i q j + m j q i ) +I (n i p j + n j p i ) + I (n i q j + n j q i ) + M (p i q j + p j q i ), L j p i = l i p j m i v j n i u j + q iw j (H + I + M ) m i m j + I n i n j + H p i p j +M q i q j + K (m i n j + m j n i ) + K(m i p j + m j p i ) + N (m i q j + m j q i ) +J (n i p j + n j p i ) + M (n i q j + n j q i ) + K (p i q j + p j q i ), L j q i = l i q j m i w j n i v j p iw j (H + I + M ) m i m j + I n i n j + K p i p j +H q i q j + N(m i n j + m j n i ) + N (m i p j + m j p i ) + M(m i q j + m j q i ) +M (n i p j + n j p i ) + J (n i q j + n j q i ) + M (p i q j + p j q i ), (2.14) L jl i = m i m j + n i n j + p i p j + q i q j, L j mi = l i m j + n i u j + p i v j + q i w j Hm i m j In i n j Kp i p j Mq i q j +(J + J + J )(m i n j + m j n i ) + (H + I + M )(m i p j + m j p i ) +(H + I + K )(m i q j + m j q i ) K (n i p j + n j p i ) N(n i q j + n j q i ) N (p i q j + p j q i ), L j ni = l i n j m i u j + p i u j + qi v j + (J + J + J )m i m j Jn i n j J p i p j J q i q j I(m i n j + m j n i ) K (m i p j + m j p i ) N(m i q j + m j q i ) I (n i p j + n j p i ) I (n i q j + n j q i ) M (p i q j + p j q i ), L j pi = l i p j m i v j n i u j + qi w j + (H + I + M ) m i m j I n i n j H p i p j M q i q j K (m i n j + m j n i ) K(m i p j + m j p i ) N (m i q j + m j q i ) J (n i p j + n j p i ) M (n i q j + n j q i ) K (p i q j + p j q i ), L j qi = l i q j m i w j n i v j pi w j + (H + I + M ) m i m j I n i n j K p i p j H q i q j N(m i n j + m j n i ) + N (m i p j + m j p i ) M(m i q j + m j q i ) + M (n i p j + n j p i ) J (n i q j + n j q i ) M (p i q j + p j q i ),

6 84 Gauree Shanker The h scalar derivative of the adapted components T αβ of the tensor Tj i is defined as [4] of (1, 1) type (2.15) T αβ,γ = (δ k T αβ ) e k γ) + T µβh µ)αγ + T αµ H µ)βγ, where δ k = k G r k r.similarly, the v-scalar derivative of the adapted components T αβ of the tensor Tj i of (1, 1) type is defined as [4] (2.16) T αβ;γ = L ( kt αβ ) e k γ) + T µβv µ)αγ + T αµ V µ)βγ. Thus, T αβ,γ and T αβ;γ are the adapted components of T i j k and T i j krespectively i. e., (2.17) (2.18) T i j k = T αβ,γ e i (α) e (β)je (γ)k, LT i j k = T αβ;γ e i (α) e (β)je (γ)k A covariant vector field σ i is called a gradient vector, if there exists a scalar field σ = σ (x) satisfying σ i = i σ. Then, we have Lemma 2.5. A covariant vector field σ i = σ α e α)i is locally a gradient vector, if and only if the scalar components σ α, α = 1, 2, 3, 4, 5. (2.19) σ α,β = σ β,α, α, β = 1, 2, 3, 4, (2.20) σ 1;α = σ α;1 = 0, α = 1, 2, 3, 4, 5 σ 2;2 = σ 2 H + σ 3 (J + J + J ) + σ 4 (H + I + M ) +σ 5 (H + I + K ), σ 3;3 = σ 2 I σ 3 J σ 4 I σ 5 I, σ 4;4 = σ 2 K σ 3 J σ 4 H σ 5 K, σ 5;5 = σ 2 M σ 3 J σ 4 M σ 5 H, σ 2;3 = σ 3;2 = σ 2 (J + J + J ) σ 3 I σ 4 K σ 5 N, σ 2;4 = σ 4;2 = σ 2 (H + I + M ) σ 3 K σ 4 K σ 5 N, σ 2;5 = σ 5;2 = σ 2 (H + I + K ) σ 3 N σ 4 N σ 5 M, σ 3;4 = σ 4;3 = σ 2 K σ 3 I σ 4 J σ 5 M, σ 3;5 = σ 5;3 = σ 2 N σ 3 I σ 4 M σ 5 J, σ 4;5 = σ 5;4 = σ 2 N σ 3 M σ 4 K σ 5 M. 97 Proof. It is obvious that σ i is locally a gradient vector if and only if it satisfies 98 (a) j σ i i σ j = 0, (b) j σ i = 0. These are equivalent, respectively, to (2.10) ( σ i j = j σ i σ k Fij) k = σj i, (2.11) σ i j = σ k Cij k 99. We examine the scalar components σ α of σ i. Then equations (2.17) and (2.18) give σ i j = σ α,β e α)i e β)j, σ i j = σ α;β e α)i e β)j respectively. Then the equations (2.10) and (2.11) are written, respectively, in the forms (2.19) and σ α;β = σ γ C αβγ. This equation together with (2.6) gives (2.20)

7 On the conformal change of five-dimensional Finsler spaces Conformal change of Cartan s connection We consider a conformal change L(x, y) L(x, y) = e σ(x) L(x, y) of a five-dimensional Finsler space F 5 = (M 5, L(x, y)) with the fundamental function L(x, y), where σ(x) is a scalar function of position x i alone, called the conformal factor. We shall denote the Finsler space with changed fundamental function L(x, y) by F 5 = (M 5, L(x, y))) and quantities of F 5 by upper line. The following change of important quantities are known [1]. (3.1) (3.2) (3.3) l i = e σ l i, m i = e σ m i, n i = e σ n i, p i = e σ p i, q i = e σ q i, g ij = e 2σ g ij, l i = e σ l i, m i = e σ m i, n i = e σ n i, p i = e σ p i, q i = e σ q i, g ij = e 2σ g ij, C ijk = e 2σ C ijk, C i jk = Cjk, i H = H, I = I, J = J, K = K, M = M, N = N, H = H, I = I, J = J, K = K, M = M, N = N, H = H, I = I, J = J, K = K, M = M Lemma (2.5) leads us to the following useful relations: Proposition 3.1. If we put σ i = i σ(x) = σ α e α)i, then we have the relations (i) σ α,β = σ β,α, α, β = 1, 2, 3, 4, 5. (ii) σ 1;α = σ α;1 = 0, α = 1, 2, 3, 4, 5. σ 2;2 = σ 1 + σ 6, σ 2;3 = σ 3;2 = σ 7, σ 2;4 = σ 4;2 = σ 8, σ 2;5 = σ 5;2 = σ 9, σ 3;3 = σ 1 + σ 10, σ 3;4 = σ 4;3 = σ 11, σ 3;5 = σ 5;3 = σ 12, σ 4;4 = σ 1 + σ 13, σ 4;5 = σ 5;4 = σ 14, σ 5;5 = σ 1 + σ 15, where we have substituted σ 6 = σ 1 σ 2 H + σ 3 (J + J + J ) + σ 4 (H + I + M ) + σ 5 (H + I + K ), σ 7 = σ 2 (J + J + J ) σ 3 I σ 4 K σ 5 N, σ 8 = σ 2 (H + I + M ) σ 3 K σ 4 K σ 5 N, σ 9 = σ 2 (H + I + K ) σ 3 N σ 4 N σ 5 M, σ 10 = σ 1 σ 2 I σ 3 J σ 4 I σ 5 I, σ 11 = σ 2 K σ 3 I σ 4 J σ 5 M, σ 12 = σ 2 N σ 3 I σ 4 M σ 5 J, σ 13 = σ 1 σ 2 K σ 3 J σ 4 H σ 5 K, σ 14 = σ 2 N σ 3 M σ 4 K σ 5 M, σ 15 = σ 1 σ 2 M σ 3 J σ 4 M σ 5 H.

8 86 Gauree Shanker Now we consider the change of Christoffel symbols γ ijk = g jr γ r ik = 1 2 ( kg ij + i g jk j g ki ) constructed from g ij (x, y) with respect to x i, then we have (3.4) γ i jk = γ i jk + δ i jσ k + δ i kσ j g jk σ i, ( σ i = g ij σ j ). Thus the change of the well-known quantities 2G i = γ i jk yj y k = γ i 00 is given by (3.5) 2G i = 2G i + L 2 ( σ 1 l i σ 2 m i σ 3 n i σ 4 p i σ 5 q i). Differentiating (3.5) with respect to y j, using proposition (3.1) and equation (2.14), we get (3.6) G i j = G i j + Ll i (σ 1 l j + σ 2 m j + σ 3 n j + σ 4 p j + σ 5 q j ) Lm i 130 (σ 2 l j + σ 6 m j + σ 7 n j + σ 8 p j + σ 9 q j ) Ln i 131 (σ 3 l j + σ 7 m j + σ 10 n j + σ 11 p j + σ 12 q j ) Lp i 132 (σ 4 l j + σ 8 m j + σ 11 n j + σ 13 p j + σ 14 q j ) Lq i 133 (σ 5 l j + σ 9 m j + σ 12 n j + σ 14 p j + σ 15 q j ). On the other hand, the connection coefficients Fjk i 134 of CΓ are given by [4] F ijk = g jr Fik r = γ ijk C ijr G r k C jkrg r i + C ikrg r 135 j. Then the equations (2.6), (3.4) and (3.6) lead to (3.7) +l i m i n i p i q i F i jk = F i jk σ 1 l j l k + σ 2 (l j m k + l k m j ) + σ 3 (l j n k + l k n j ) + σ 4 (l j p k + l k p j ) + σ 5 (l j q k + l k q j ) +σ 6 m j m k + σ 7 (m j n k + m k n j ) + σ 8 (m j p k + m k p j ) + σ 9 (m j q k + m k q j ) + σ 10 n j n k +σ 11 (n j p k + n k p j ) + σ 12 (n j q k + n k q j ) + σ 13 p j p k + σ 14 (p j q k + p k q j ) + σ 15 q j q k σ 2 l j l k + σ 6 (l j m k + l k m j ) + σ 7 (l j n k + l k n j ) + σ 8 (l j p k + l k p j ) + σ 9 (l j q k + l k q j ) +σ 16 m j m k + σ 17 (m j n k + m k n j ) + σ 18 (m j p k + m k p j ) + σ 19 (m j q k + m k q j ) + σ 20 n j n k +σ 21 (n j p k + n k p j ) + σ 22 (n j q k + n k q j ) + σ 23 p j p k + σ 24 (p j q k + p k q j ) + σ 25 q j q k σ 3 l j l k + σ 7 (l j m k + l k m j ) + σ 10 (l j n k + l k n j ) + σ 11 (l j p k + l k p j ) + σ 12 (l j q k + l k q j ) +σ 26 m j m k + σ 27 (m j n k + m k n j ) + σ 28 (m j p k + m k p j ) + σ 29 (m j q k + m k q j ) + σ 30 n j n k + σ 31 (n j p k + n k p j ) + σ 32 (n j q k + n k q j ) + σ 33 p j p k + σ 34 (p j q k + p k q j ) + σ 35 q j q k σ 4 l j l k + σ 8 (l j m k + l k m j ) + σ 11 (l j n k + l k n j ) + σ 13 (l j p k + l k p j ) + σ 14 (l j q k + l k q j ) +σ 36 m j m k + σ 37 (m j n k + m k n j ) + σ 38 (m j p k + m k p j ) + σ 39 (m j q k + m k q j ) + σ 40 n j n k +σ 41 (n j p k + n k p j ) + σ 42 (n j q k + n k q j ) + σ 43 p j p k + σ 44 (p j q k + p k q j ) + σ 45 q j q k σ 5 l j l k + σ 9 (l j m k + l k m j ) + σ 12 (l j n k + l k n j ) + σ 14 (l j p k + l k p j ) + σ 15 (l j q k + l k q j ) +σ 46 m j m k + σ 47 (m j n k + m k n j ) + σ 48 (m j p k + m k p j ) + σ 49 (m j q k + m k q j ) + σ 50 n j n k +σ 51 (n j p k + n k p j ) + σ 52 (n j q k + n k q j ) + σ 53 p j p k + σ 54 (p j q k + p k q j ) + σ 55 q j q k where σ 16 = σ 2 σ 6 H + σ 7 (J + J + J ) + σ 8 (H + I + M ) + σ 9 (H + I + K ), σ 17 = σ 3 σ 7 H + σ 10 (J + J + J ) + σ 11 (H + I + M ) + σ 12 (H + I + K ), σ 18 = σ 4 σ 8 H + σ 11 (J + J + J ) + σ 13 (H + I + M ) + σ 14 (H + I + K ), σ 19 = σ 5 σ 9 H + σ 12 (J + J + J ) + σ 14 (H + I + M ) + σ 15 (H + I + K ), σ 20 = σ 2 + σ 6 I + σ 7 (3J + 2J + 2J ) + σ 8 I + σ 9 I 2σ 10 I 2σ 11 K 2σ 12 N, σ 21 = σ 6 K + σ 7 (H + 2I + M ) + σ 8 (J + 2J + J ) + σ 9 M σ 10 K σ 11 (K + I) σ 12 N σ 13 K σ 14 N,,

9 On the conformal change of five-dimensional Finsler spaces σ 22 = σ 6 N + σ 7 (H + 2I + K ) + σ 8 M + σ 9 (J + J + 2J ) σ 10 N σ 11 N σ 12 (I + M) σ 14 K σ 15 N, σ 23 = σ 2 + σ 6 K + σ 7 J + σ 8 (3H + 2I + 2M ) + σ 9 K 2 (σ 11 K + σ 13 K + σ 14 N ), σ 24 = σ 6 N + σ 7 M + σ 8 (H + I + K ) + σ 9 (H + I + 2M ) σ 11 N σ 12 K σ 13 N σ 14 (M + K), σ 25 = σ 2 +σ 6 M+σ 7 J +σ 8 M +σ 9 (3H + 2I + 2K ) 2 (σ 12 N + σ 14 N + σ 15 M), σ 26 = σ 3 + 2σ 6 (J + J + J ) σ 7 (H + 2I) 2 (σ 8 K + σ 9 N) σ 10 (J + J + J ) σ 11 (H + I + M ) σ 12 (H + I + K ), σ 27 = σ 2 σ 6 I σ 7 J σ 8 I σ 9 I, σ 28 = σ 6 K + σ 8 (J + J ) σ 7 (H + 2I + M ) σ 9 M + σ 10 K +σ 11 (K I) + σ 12 N σ 13 K σ 14 N, σ 29 = σ 6 N σ 7 (H + 2I + M ) σ 8 M + σ 9 (J + J ) + σ 10 N +σ 11 N + σ 12 (M I) σ 14 K σ 15 N, σ 30 = σ 3 σ 7 I σ 10 J σ 11 I σ 12 I, σ 31 = σ 4 σ 8 I σ 11 J σ 13 I σ 14 I, σ 32 = σ 5 σ 9 I σ 12 J σ 14 I σ 15 I, σ 33 = σ 3 + σ 7 K 2σ 8 K + σ 11 (H 2I ) + σ 12 K 2σ 13 J 2σ 14 M, σ 34 = σ 7 N σ 8 N σ 9 K + σ 10 M σ 11 (K I ) + σ 12 (M I ) σ 13 M σ 14 (J + J ) σ 15 M, σ 35 = σ 3 + σ 7 M 2σ 9 N + σ 10 J + σ 11 M + σ 12 (H 2I ) σ 14 M σ 15 J, σ 36 = σ 4 + 2σ 6 (H + I + M ) 2σ 7 K + σ 8 (H 2K) 2σ 9 N σ 11 (J + J + J ) σ 13 (H + I + M ) σ 14 (H + I + K ), σ 37 = σ 6 K + σ 7 (H + M ) σ 8 (J + 2J + J ) σ 9 M σ 10 K +σ 11 I K σ 12 N + σ 13 K + σ 14 N, σ 38 = σ 2 σ 6 K σ 7 J σ 8 H σ 9 K, σ 39 = σ 6 N σ 7 M σ 8 (H + I + 2K ) + σ 9 (H + I ) + σ 11 N σ 12 K + σ 13 N σ 14 (K M) σ 15 N, σ 40 = σ 4 2σ 7 K + σ 8 I 2σ 10 I + σ 11 (J 2J ) 2σ 12 M + σ 13 I + σ 14 I, σ 41 = σ 3 σ 7 K σ 10 J σ 11 H σ 12 K, σ 42 = σ 7 N + σ 8 N σ 9 K σ 10 M σ 11 (K I ) σ 12 (M + I ) + σ 13 M +σ 14 (J J ) σ 15 M, σ 43 = σ 4 σ 8 K σ 11 J σ 13 H σ 14 K, σ 44 = σ 5 σ 9 K σ 12 J σ 14 H σ 15 K, σ 45 = σ 4 + σ 8 M σ 9 N + σ 11 J σ 12 M + σ 13 M + σ 14 (H K ) σ 15 M, σ 46 = σ 5 +2σ 6 (H + I + K ) 2σ 7 N 2σ 8 N +σ 9 (H 2M) σ 12 (J + J + J ) σ 14 (H + I + M ) σ 15 (H + I + K ), σ 47 = σ 6 N + σ 7 (H + K ) σ 8 M σ 9 (J + J + J ) σ 10 N σ 11 N σ 12 (M I) + σ 14 K + σ 15 N, σ 48 = σ 6 N σ 7 M + σ 8 (H + I ) σ 9 (H + I + 2M ) σ 11 N +σ 12 K σ 13 N + σ 14 (K M) + σ 15 N, σ 49 = σ 2 σ 6 M σ 7 J σ 8 M σ 9 H, σ 50 = σ 5 2σ 7 N + σ 9 I 2σ 10 I 2σ 11 M + σ 12 (J 2J ) + σ 14 I + σ 15 I, σ 51 = σ 7 N σ 8 N + σ 9 K σ 10 M σ 11 (K + I ) σ 12 (M I ) σ 13 M σ 14 (J J ) + σ 15 M, σ 52 = σ 3 σ 7 M σ 10 J σ 11 M σ 12 H, σ 53 = σ 5 2σ 8 N + σ 9 K 2σ 11 M + σ 12 J 2σ 13 K

10 88 Gauree Shanker σ 14 (H 2M ) + σ 15 K, σ 54 = σ 4 σ 8 M σ 11 J σ 13 M σ 14 H, σ 55 = σ 5 σ 9 M σ 12 J σ 14 M σ 15 H. Now we shall deal with the conformally invariant scalar field S(x, y). Its h-covariant derivative S i with respect to the changed CΓ is defined by S i = i S ( j S) G j i. It is enough for the later use to treat a positively homogeneous scalar field S of degree zero in y i so that S ;1 = 0. Then from (3.6), we have (3.8) S i = S i + S ;2 (σ 2 l i + σ 6 m i + σ 7 n i + σ 8 p i + σ 9 q i ) S ;3 (σ 3 l i + σ 7 m i + σ 10 n i + σ 11 p i + σ 12 q i ) +S ;4 (σ 4 l i + σ 8 m i + σ 11 n i + σ 13 p i + σ 14 q i ) +S ;5 (σ 5 l i + σ 9 m i + σ 12 n i + σ 14 p i + σ 15 q i ). Since S i = S, 1 l i + S, 2 m i + S, 3 n i + S, 4 p i + S, 5 q i, from the equations (3.2) and (3.8) we have the relations: (3.9) S, 1 = S i l i = e σ (S, 1 +S ;2 σ 2 + S ;3 σ 3 + S ;4 σ 4 + S ;5 σ 5 ), S, 2 = S i m i = e σ (S, 2 +S ;2 σ 6 + S ;3 σ 7 + S ;4 σ 8 + S ;5 σ 9 ), S, 3 = S i n i = e σ (S, 3 +S ;2 σ 7 + S ;3 σ 10 + S ;4 σ 11 + S ;5 σ 12 ), S, 4 = S i p i = e σ (S, 4 +S ;2 σ 8 + S ;3 σ 11 + S ;4 σ 13 + S ;5 σ 14 ), S, 5 = S i q i = e σ (S, 5 +S ;2 σ 9 + S ;3 σ 12 + S ;4 σ 14 + S ;5 σ 15 ). On the other hand, the v-covariant derivative S i with respect to the changed CΓ is defined by S i = i S = S i. Making use of the relation (3.2), this equation gives (3.10) S ;1 = LS i l i = 0, S ;2 = LS i m i = S ;2, S ;3 = LS i n i = S ;3, S ;4 = LS i p i = S ;4, S ;5 = LS i q i = S ;5. Proposition 3.2. Let S be a conformally invariant scalar field, which is positively homogeneous of degree zero in y i. Then the conformal changes of scalar derivatives of S are given by (3.9) and (3.10). For the conformal change of the adapted components h α, J α, k α, h α, J α, k α of the six h-connection vectors h i, J i, k i, h i, J i, k i, from (3.1) and (2.10), we have m i j = e σ (σ j m i + m i j ), n i j = e σ (σ j n i + n i j ), p i j = e σ (σ j p i + p i j ), q i j = e σ (σ j q i + q i j ) which in view of (3.6) and (3.7) leads to (3.11) (a) h j = h j + {σ 2 u 2 + σ 3 u 3 + σ 4 u 4 + σ 5 u 5 } l j + {σ 6 u 2 + σ 7 u 3 + σ 8 u 4 + σ 9 u 5 + σ 17 + σ 56 } m j + {σ 7 u 2 + σ 10 u 3 + σ 11 u 4 + σ 12 u 5 + σ 20 + σ 57 } n j + {σ 8 u 2 + σ 11 u 3 + σ 13 u 4 + σ 14 u 5 + σ 21 + σ 58 } p j + {σ 9 u 2 + σ 12 u 3 + σ 14 u 4 + σ 15 u 5 + σ 22 + σ 59 } q j.

11 On the conformal change of five-dimensional Finsler spaces (b) J j = J j + {σ 2 v 2 + σ 3 v 3 + σ 4 v 4 + σ 5 v 5 } l j + {σ 6 v 2 + σ 7 v 3 + σ 8 v 4 + σ 9 v 5 + σ 18 + σ 60 } m j + {σ 7 v 2 + σ 10 v 3 + σ 11 v 4 + σ 12 v 5 + σ 21 + σ 61 } n j + {σ 8 v 2 + σ 11 v 3 + σ 13 v 4 + σ 14 v 5 + σ 23 + σ 62 } p j, + {σ 9 v 2 + σ 12 v 3 + σ 14 v 4 + σ 15 v 5 + σ 24 + σ 63 } q j. (c) k j = k j + {σ 2 w 2 + σ 3 w 3 + σ 4 w 4 + σ 5 w 5 } l j + {σ 6 w 2 + σ 7 w 3 + σ 8 w 4 + σ 9 w 5 + σ 19 + σ 64 } m j + {σ 7 w 2 + σ 10 w 3 + σ 11 w 4 + σ 12 w 5 + σ 22 + σ 65 } n j + {σ 8 w 2 + σ 11 w 3 + σ 13 w 4 + σ 14 w 5 + σ 24 + σ 66 } p j + {σ 9 w 2 + σ 12 w 3 + σ 14 w 4 + σ 15 w 5 + σ 25 + σ 67 } q j. (d) h j = h j + {σ 2u 2 + σ 3 u 3 + σ 4 u 4 + σ 5 u 5} l j + {σ 6 u 2 + σ 7 u 3 + σ 8 u 4 + σ 9 u 5 + σ 28 + σ 68 } m j + {σ 7 u 2 + σ 10 u 3 + σ 11 u 4 + σ 12 u 5 + σ 31 + σ 69 } n j + {σ 8 u 2 + σ 11 u 3 + σ 13 u 4 + σ 14 u 5 + σ 33 + σ 70 } p j + {σ 9 u 2 + σ 12 u 3 + σ 14 u 4 + σ 15 u 5 + σ 34 + σ 71 } q j. (e) J j = J j + {σ 2v 2 + σ 3 v 3 + σ 4 v 4 + σ 5 v 5} l j + {σ 6 v 2 + σ 7 v 3 + σ 8 v 4 + σ 9 v 5 + σ 29 + σ 72 } m j + {σ 7 v 2 + σ 10 v 3 + σ 11 v 4 + σ 12 v 5 + σ 32 + σ 73 } n j + {σ 8 v 2 + σ 11 v 3 + σ 13 v 4 + σ 14 v 5 + σ 34 + σ 74 } p j, + {σ 9 v 2 + σ 12 v 3 + σ 14 v 4 + σ 15 v 5 + σ 35 + σ 75 } q j. (f) k j = k j + {σ 2w 2 + σ 3 w 3 + σ 4 w 4 + σ 5 w 5} l j + {σ 6 w 2 + σ 7 w 3 + σ 8 w 4 + σ 9 w 5 + σ 39 + σ 76 } m j + {σ 7 w 2 + σ 10 w 3 + σ 11 w 4 + σ 12 w 5 + σ 42 + σ 77 } n j + {σ 8 w 2 + σ 11 w 3 + σ 13 w 4 + σ 14 w 5 + σ 44 + σ 78 } p j + {σ 9 w 2 + σ 12 w 3 + σ 14 w 4 + σ 15 w 5 + σ 45 + σ 79 } q j. where we have put σ 56 = σ 6 (J + J + J ) + σ 7 I + σ 8 K + σ 9 N, σ 57 = σ 7 (J + J + J ) + σ 10 I + σ 11 K + σ 12 N, σ 58 = σ 8 (J + J + J ) + σ 11 I + σ 13 K + σ 14 N, σ 59 = σ 9 (J + J + J ) + σ 12 I + σ 14 K + σ 15 N, σ 60 = σ 6 (H + I + M ) + σ 7 K + σ 8 K + σ 9 N, σ 61 = σ 7 (H + I + M ) + σ 10 K + σ 11 K + σ 12 N, σ 62 = σ 8 (H + I + M ) + σ 11 K + σ 13 K + σ 14 N, σ 63 = σ 9 (H + I + M ) + σ 12 K + σ 14 K + σ 15 N, σ 64 = σ 6 (H + I + K ) + σ 7 N + σ 8 N + σ 9 M, σ 65 = σ 7 (H + I + K ) + σ 10 N + σ 11 N + σ 12 M, σ 66 = σ 8 (H + I + K ) + σ 11 N + σ 13 N + σ 14 M, σ 67 = σ 9 (H + I + K ) + σ 12 N + σ 14 N + σ 15 M, σ 68 = σ 6 K + σ 7 I + σ 8 J + σ 9 M, σ 69 = σ 7 K + σ 10 I + σ 11 J + σ 12 M, σ 70 = σ 8 K + σ 11 I + σ 13 J + σ 14 M, σ 71 = σ 9 K + σ 12 I + σ 14 J + σ 15 M, σ 72 = σ 6 N + σ 7 I + σ 8 M + σ 9 J, σ 73 = σ 7 N + σ 10 I + σ 11 M + σ 12 J, σ 74 = σ 8 N + σ 11 I + σ 13 M + σ 14 J, σ 75 = σ 9 N + σ 12 I + σ 14 M + σ 15 J, σ 76 = σ 6 N + σ 7 M + σ 8 K + σ 9 M,

12 90 Gauree Shanker σ 77 = σ 7 N + σ 10 M + σ 11 K + σ 12 M, σ 78 = σ 8 N + σ 11 M + σ 13 K + σ 14 M, σ 79 = σ 9 N + σ 12 M + σ 14 K + σ 15 M. ( Thus the adapted components h α, J α, k α, h α, J α, k α, of h i, J i, k i M 5, L(x, y) ) are given by in F 5 = (3.12) (a) h 1 = e σ {h 1 + σ 2 u 2 + σ 3 u 3 + σ 4 u 4 + σ 5 u 5 }, h 2 = e σ {h 2 + σ 6 u 2 + σ 7 u 3 + σ 8 u 4 + σ 9 u 5 + σ 17 + σ 56 }, h 3 = e σ {h 3 + σ 7 u 2 + σ 10 u 3 + σ 11 u 4 + σ 12 u 5 + σ 20 + σ 57 }, h 4 = e σ {h 4 + σ 8 u 2 + σ 11 u 3 + σ 13 u 4 + σ 14 u 5 + σ 21 + σ 58 }, h 5 = e σ {h 5 + σ 9 u 2 + σ 12 u 3 + σ 14 u 4 + σ 15 u 5 + σ 22 + σ 59 } (b) J 1 = e σ {J 1 + σ 2 v 2 + σ 3 v 3 + σ 4 v 4 + σ 5 v 5 }, J 2 = e σ {J 2 + σ 6 v 2 + σ 7 v 3 + σ 8 v 4 + σ 9 v 5 + σ 18 + σ 60 }, J 3 = e σ {J 3 + σ 7 v 2 + σ 10 v 3 + σ 11 v 4 + σ 12 v 5 + σ 21 + σ 61 }, J 4 = e σ {J 4 + σ 8 v 2 + σ 11 v 3 + σ 13 v 4 + σ 14 v 5 + σ 23 + σ 62 }, J 5 = e σ {J 5 + σ 9 v 2 + σ 12 v 3 + σ 14 v 4 + σ 15 v 5 + σ 24 + σ 63 }. (c) k 1 = e σ {k 1 + σ 2 w 2 + σ 3 w 3 + σ 4 w 4 + σ 5 w 5 }, k 2 = e σ {k 2 + σ 6 w 2 + σ 7 w 3 + σ 8 w 4 + σ 9 w 5 + σ 19 + σ 64 }, k 3 = e σ {k 3 + σ 7 w 2 + σ 10 w 3 + σ 11 w 4 + σ 12 w 5 + σ 22 + σ 65 }, k 4 = e σ {k 4 + σ 8 w 2 + σ 11 w 3 + σ 13 w 4 + σ 14 w 5 + σ 24 + σ 66 }, k 5 = e σ {k 5 + σ 9 w 2 + σ 12 w 3 + σ 14 w 4 + σ 15 w 5 + σ 25 + σ 67 }. (d) h 1 = e σ {h 1 + σ 2 u 2 + σ 3 u 3 + σ 4 u 4 + σ 5 u 5}, h 2 = e σ {h 2 + σ 6 u 2 + σ 7 u 3 + σ 8 u 4 + σ 9 u 5 + σ 28 + σ 68 }, h 3 = e σ {h 3 + σ 7 u 2 + σ 10 u 3 + σ 11 u 4 + σ 12 u 5 + σ 31 + σ 69 }, h 4 = e σ {h 4 + σ 8 u 2 + σ 11 u 3 + σ 13 u 4 + σ 14 u 5 + σ 33 + σ 70 }, h 5 = e σ {h 5 + σ 9 u 2 + σ 12 u 3 + σ 14 u 4 + σ 15 u 5 + σ 34 + σ 71 }. (e) J 1 = e σ {J 1 + σ 2 v 2 + σ 3 v 3 + σ 4 v 4 + σ 5 v 5}, J 2 = e σ {J 2 + σ 6 v 2 + σ 7 v 3 + σ 8 v 4 + σ 9 v 5 + σ 29 + σ 72 }, J 3 = e σ {J 3 + σ 7 v 2 + σ 10 v 3 + σ 11 v 4 + σ 12 v 5 + σ 32 + σ 73 }, J 4 = e σ {J 4 + σ 8 v 2 + σ 11 v 3 + σ 13 v 4 + σ 14 v 5 + σ 34 + σ 74 }, J 5 = e σ {J 5 + σ 9 v 2 + σ 12 v 3 + σ 14 v 4 + σ 15 v 5 + σ 35 + σ 75 }. (f) k 1 = e σ {k 1 + σ 2 w 2 + σ 3 w 3 + σ 4 w 4 + σ 5 w 5}, k 2 = e σ {k 2 + σ 6 w 2 + σ 7 w 3 + σ 8 w 4 + σ 9 w 5 + σ 39 + σ 76 }, k 3 = e σ {k 3 + σ 7 w 2 + σ 10 w 3 + σ 11 w 4 + σ 12 w 5 + σ 42 + σ 77 }, k 4 = e σ {k 4 + σ 8 w 2 + σ 11 w 3 + σ 13 w 4 + σ 14 w 5 + σ 44 + σ 78 }, k 5 = e σ {k 5 + σ 9 w 2 + σ 12 w 3 + σ 14 w 4 + σ 15 w 5 + σ 45 + σ 79 }. For the conformal change of the adapted components u α, v α, w α, u α, v α, w α, of six v-connection vectors u i, v i, w i, u i, v i, w i, we again use (3.1) and (2.11). Thus we get u j = e σ u j, v j = e σ v j, w j = e σ w j, u j = e σ u j, v j = e σ v j, w j = e σ w j, which lead to (3.13) (a) u 1 = u 1 = 0, u 2 = u 2, u 3 = u 3, u 4 = u 4, u 5 = u 5, (b) v 1 = v 1 = 0, v 2 = v 2, v 3 = v 3, v 4 = v 4, v 5 = v 5, (c) w 1 = w 1 = 0, w 2 = w 2, w 3 = w 3, w 4 = w 4, w 5 = w 5, (d) u 1 = u 1 = 0, u 2 = u 2, u 3 = u 3, u 4 = u 4, u 5 = u 5,

13 On the conformal change of five-dimensional Finsler spaces (e) v 1 = v 1 = 0, v 2 = v 2, v 3 = v 3, v 4 = v 4, v 5 = v 5, (f) w 1 = w 1 = 0, w 2 = w 2, w 3 = w 3, w 4 = w 4, w 5 = w 5. From (3.11) and (3.13), we have the following: Theorem 3.3. The adapted components of all the six v-connection vectors of fivedimensional Finsler space are invariant under any conformal change. Theorem 3.4. The h-connection vector h i of F 5 is invariant under σ-conformal change if and only if (i) σ 2 u 2 + σ 3 u 3 + σ 4 u 4 + σ 5 u 5 = 0, (ii) σ 6 u 2 + σ 7 u 3 + σ 8 u 4 + σ 9 u 5 + σ 17 + σ 56 = 0, (iii) σ 7 u 2 + σ 10 u 3 + σ 11 u 4 + σ 12 u 5 + σ 20 + σ 57 = 0, (iv) σ 8 u 2 + σ 11 u 3 + σ 13 u 4 + σ 14 u 5 + σ 21 + σ 58 = 0, (v) σ 9 u 2 + σ 12 u 3 + σ 14 u 4 + σ 15 u 5 + σ 22 + σ 59 = 0. Theorem 3.5. The h-connection vector J i of F 5 is invariant under σ-conformal change if and only if (i) σ 2 v 2 + σ 3 v 3 + σ 4 v 4 + σ 5 v 5 = 0, (ii) σ 6 v 2 + σ 7 v 3 + σ 8 v 4 + σ 9 v 5 + σ 18 + σ 60 = 0, (iii) σ 7 v 2 + σ 10 v 3 + σ 11 v 4 + σ 12 v 5 + σ 21 + σ 61 = 0, (iv) σ 8 v 2 + σ 11 v 3 + σ 13 v 4 + σ 14 v 5 + σ 23 + σ 62 = 0, (v) σ 9 v 2 + σ 12 v 3 + σ 14 v 4 + σ 15 v 5 + σ 24 + σ 63 = 0. Theorem 3.6. The h-connection vector k i of F 5 is invariant under σ-conformal change if and only if (i) σ 2 w 2 + σ 3 w 3 + σ 4 w 4 + σ 5 w 5 = 0, (ii) σ 6 w 2 + σ 7 w 3 + σ 8 w 4 + σ 9 w 5 + σ 19 + σ 64 = 0, (iii) σ 7 w 2 + σ 10 w 3 + σ 11 w 4 + σ 12 w 5 + σ 22 + σ 65 = 0, (iv) σ 8 w 2 + σ 11 w 3 + σ 13 w 4 + σ 14 w 5 + σ 24 + σ 66 = 0, (v) σ 9 w 2 + σ 12 w 3 + σ 14 w 4 + σ 15 w 5 + σ 25 + σ 67 = 0. Theorem 3.7. The h-connection vector h i of F 5 is invariant under σ-conformal change if and only if (i) σ 2 u 2 + σ 3 u 3 + σ 4 u 4 + σ 5 u 5 = 0, (ii) σ 6 u 2 + σ 7 u 3 + σ 8 u 4 + σ 9 u 5 + σ 28 + σ 68 = 0, (iii) σ 7 u 2 + σ 10 u 3 + σ 11 u 4 + σ 12 u 5 + σ 31 + σ 69 = 0, (iv) σ 8 u 2 + σ 11 u 3 + σ 13 u 4 + σ 14 u 5 + σ 33 + σ 70 = 0, (v) σ 9 u 2 + σ 12 u 3 + σ 14 u 4 + σ 15 u 5 + σ 34 + σ 71 = 0. Theorem 3.8. The h-connection vector J i of F 5 is invariant under σ-conformal change if and only if (i) σ 2 v 2 + σ 3 v 3 + σ 4 v 4 + σ 5 v 5 = 0, (ii) σ 6 v 2 + σ 7 v 3 + σ 8 v 4 + σ 9 v 5 + σ 29 + σ 72 = 0, (iii) σ 7 v 2 + σ 10 v 3 + σ 11 v 4 + σ 12 v 5 + σ 32 + σ 73 = 0, (iv) σ 8 v 2 + σ 11 v 3 + σ 13 v 4 + σ 14 v 5 + σ 34 + σ 74 = 0, (v) σ 9 v 2 + σ 12 v 3 + σ 14 v 4 + σ 15 v 5 + σ 35 + σ 75 = 0.

14 92 Gauree Shanker Theorem 3.9. The h-connection vector k i of F 5 is invariant under σ-conformal change if and only if (i) σ 2 w 2 + σ 3 w 3 + σ 4 w 4 + σ 5 w 5 = 0, (ii) σ 6 w 2 + σ 7 w 3 + σ 8 w 4 + σ 9 w 5 + σ 39 + σ 76 = 0, (iii) σ 7 w 2 + σ 10 w 3 + σ 11 w 4 + σ 12 w 5 + σ 42 + σ 77 = 0, (iv) σ 8 w 2 + σ 11 w 3 + σ 13 w 4 + σ 14 w 5 + σ 44 + σ 78 = 0, (v) σ 9 w 2 + σ 12 w 3 + σ 14 w 4 + σ 15 w 5 + σ 45 + σ 79 = References [1] M. Hashiguchi, On conformal transformations of Finsler metrics, J. Math. Kyoto Univ. 16 (1976), [2] M. Matsumoto, Conformal change of two-dimensional Finsler space and curvature of one-form metric, Tensor N. S. 53 (1993), [3] M. Matsumoto, A theory of three-dimensional Finsler spaces in terms of scalars, Demonstratio Mathematica 6 (1973), [4] M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Saikawa, Otsu, 520, Japan, [5] M. Matsumoto and R. Miron, On an invariant theory of Finsler spaces, Period. Math. Hungar. 8 (1977), [6] A. Moor, Über. die Torsions-Und Krümmung invarianten der dreidimensonalen 356 Finslerschen Räume, Math. Nachr. 16 (1957), [7] T. N. Pandey and D. K. Divedi, A theory of four-dimensional Finsler spaces in 358 terms of scalars, J. Nat. Acad. Math. 11 (1997), [8] B. N. Prasad and D. K. Diwedi, Conformal change of three-dimensional Finsler 360 space, Tensor N. S. 61 (1999), [9] B. N. Prasad, G. C. Chaubey and G. S. Patel, The four-dimensional Finsler 362 space with constant unified main scalar, Bull. Calcutta Math. Soc. 99, 2 (2007), [10] B. N. Prasad and G. Shanker, Conformal change of four-dimensional Finsler 365 space, Bull. Calcutta Math. Soc. 102, 5 (2010), [11] G. Shanker, G. C. Chaubey and V. Pandey, On the main scalars of a five- 367 dimensional Finsler space, Int. J. Pure Appl. Math. 5, 2 (2012), [12] G. Shanker, Five dimensional Finsler space with constant unified main scalar, 369 Tensor N. S. 72, 1 (2010), [13] U. P. Singh and B. Kumari, Conformal change of three-dimensional Finsler space 371 with constant unified main scalars, J. Pur. Acad. Sci. 6 (2000), Author s address: Gauree Shanker Department of Mathematics and Statistics, Banasthali University, Banasthali, Rajasthan , India. grshnkr2007@gmail.com