On the conformal change of five-dimensional Finsler spaces Gauree Shanker 1 2 3 4 5 6 7 8 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 20 21 22 23 19 ( 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 24 25 26 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. 79-92. c Balkan Society of Geometers, Geometry Balkan Press 2013.
80 Gauree Shanker 27 28 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
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 ), 40 41 42 43 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 ) 44 45 46 47 48 49 +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 = 0 0 0 0 0 0 0 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,
82 Gauree Shanker 50 51 52 53 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 = 0. 63 64 65 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
On the conformal change of five-dimensional Finsler spaces 83 67 68 69 70 71 72 73 74 75 76 77 78 (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, 79 80 81 82 83 84 85 86 87 88 89 90 91 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 ),
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. 92 93 94 95 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, 5. 96 (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). 100 101 102 103
On the conformal change of five-dimensional Finsler spaces 85 104 3 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. 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 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.
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 136 137 138 139 140 141 142 143 144 145 146 147 148 149 (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,,
On the conformal change of five-dimensional Finsler spaces 87 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 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
88 Gauree Shanker 197 198 199 200 201 202 +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 ) 203 204 205 +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 ), 206 207 208 209 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, 210 211 212 213 214 215 216 217 218 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.
On the conformal change of five-dimensional Finsler spaces 89 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 (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,
90 Gauree Shanker 266 267 268 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 }. 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 (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, 294 295 296 (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,
On the conformal change of five-dimensional Finsler spaces 91 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 (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.
92 Gauree Shanker 337 338 339 340 341 342 343 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 = 0. 344 345 346 347 348 349 350 351 352 353 354 References [1] M. Hashiguchi, On conformal transformations of Finsler metrics, J. Math. Kyoto Univ. 16 (1976), 25-50. [2] M. Matsumoto, Conformal change of two-dimensional Finsler space and curvature of one-form metric, Tensor N. S. 53 (1993), 149-161. [3] M. Matsumoto, A theory of three-dimensional Finsler spaces in terms of scalars, Demonstratio Mathematica 6 (1973), 1-29. [4] M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Saikawa, Otsu, 520, Japan, 1986. [5] M. Matsumoto and R. Miron, On an invariant theory of Finsler spaces, Period. Math. Hungar. 8 (1977), 73-82. 355 [6] A. Moor, Über. die Torsions-Und Krümmung invarianten der dreidimensonalen 356 Finslerschen Räume, Math. Nachr. 16 (1957), 85-99. 357 [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), 176-190. 359 [8] B. N. Prasad and D. K. Diwedi, Conformal change of three-dimensional Finsler 360 space, Tensor N. S. 61 (1999), 148-157. 361 [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), 363 113-122. 364 [10] B. N. Prasad and G. Shanker, Conformal change of four-dimensional Finsler 365 space, Bull. Calcutta Math. Soc. 102, 5 (2010), 423-432. 366 [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), 69-78. 368 [12] G. Shanker, Five dimensional Finsler space with constant unified main scalar, 369 Tensor N. S. 72, 1 (2010), 79-85. 370 [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), 1-13. 372 373 374 375 376 377 Author s address: Gauree Shanker Department of Mathematics and Statistics, Banasthali University, Banasthali, Rajasthan-304022, India. E-mail: grshnkr2007@gmail.com