ON h-transformation OF SOME SPECIAL FINSLER SPACE

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1 Electonc Jounal of Mathematcal Analyss and Applcatons Vol. 5(2) July 2017, pp ISSN: X(onlne) ON h-transformation OF SOME SPECIAL FINSLER SPACE MANOJ KUMAR SINGH AND SAMIDHA YADAV Abstact. The pupose of the pesent pape s to fnd the elaton between the v-cuvatue tenso wth espect to Catan s connecton of Fnsle space F n = (M n, L) and F n = (M n, L) whee L(x, y) s obtaned fom L(x, y) by the tansfomaton L(x, y) = e σ L(x, y) + b (x, y)y and b (x, y) s an h- vecto n (M n, L). we shall also study the popetes of Fnsle space F n unde the condton that F n s some specal Fnsle space. In patcula of e σ L(x, y) s confomal change then (v)h and (v)hv toson tensos of (M n, L) have been obtaned. 1. Intoducton Let F n = (M n, L) be an n-dmensonal Fnsle space, whee M n s an n- dmensonal dffeentable manfold and L(x, y) s the Fnsle fundamental, functon. Matsumoto [1] ntoduced tansfomaton of Fnsle metc L = e σ L + b (x)y (1.1) and obtaned the elaton between the Catan s connecton coeffcents of F n and F n =(M n, L). It has been assumed that the functon b n (1.1) ae functons of co- odnates x only. If n (1.1) σ(x) vanshes and L(x, y) s a metc functon of Remannan space then L(x, y) educes to the Randes Space whch s ntoduced by G. Randes [3]. If L(x, y) s a metc functon of Remannan space then L(x, y) educes to the β-confomal change. H. Izum [2] ntoduced the h-vecto b (x, y) n the confomal tansfomaton of Fnsle space, whch s v-covaantly constant wth espect to Catan s connecton CΓ and satsfes LC h j b n =ρh j whee C h j s Catan s C-tenso, h j s the angula metc tenso, ρ a functon whch depends only on co-odnates and s gven by, ρ = 1 (n 1) L C b and C = C jk gjk s the toson vecto. Thus the h-vecto b s not only a functon of co-odnates but t s a functon of dectonal aguement satsfyng L b y = ρ h j. Many authos A.Taleshan et.al.[10] and S.H. Abed [11] studed the popetes of such Fnsle Spaces obtaned by ths metc. In ths pape we consde the metc functon gven by equaton L = e σ L(x, y) + b (x, y)y, whch genealzes many Changes n Fnsle geomety, called h- confomal tansfofmaton of Fnsle metc. The secton second 2010 Mathematcs Subject Classfcaton. 52B40, 53C60. Key wods and phases. Confomal tansfomaton, h-tansfomaton, (h)-toson tenso, (hv)- toson tenso, v-cuvatue tenso. Submtted Sep. 9,

2 EJMAA-2017/5(2) ON H-TRANSFORMATION OF SOME SPECIAL FINSLER 251 of ths pape gves the elaton between Catan connecton CΓ of F n =(M n, L) and F n = (M n, L). The thd secton s devoted to fnd the toson tensos R jk of F n and we consde the case that ths space s of scala cuvatue. The fouth secton s devoted to fnd the toson tenso P hjk and to consde the case that ths space becomes a Landsbeg space. Fo an h-vecto b, we have the followng[2]. Lemma 1.1 If b s an h-vecto then the functon ρ and l = b - ρe σ l ae ndependent of y. Lemma 1.2 The magntude b of an h-vecto b s ndependent of y. 2. Catan s connecton of the space F n Let b be a vecto feld n the Fnsle space (M n, L), f b satsfes the condtons (1)b j = 0 (2)LC h jb h = ρh j (2.1) then the vecto feld b s called an h vecto[2]. Hee j denote the v-covaant devatve wth espect to Catan s connecton CΓ. Cj h s the Catan s C tenso, h j s the angula metc tenso and ρ be a functon gven by ρ = (n 1) 1 LC b (2.2) whee C s the toson vecto C jkg jk. fom (2.1) we get ρ jb = L 1 ρh j (2.3) Thoughout the pape we shall use the notaton L = L, L j = j L.... The quanttes and opeatons efeeng to F n ae ndcated by puttng ba, thus fom (1.1) we get (a) L = e σ L + b (b) L j = (e σ + ρ)l j (c) L jk = (e σ + ρ)l jk (d) L jkh = (e σ + ρ)l jkh (2.4) and so on. If l, h j, g j and C jk denote the nomalzed element of suppot, the angula metc tenso, the fundamental metc tenso and Catan s C-tenso of F n espectvely, then these quanttes n F n ae obtaned by (2.4) as [9] l = e σ l + b (2.5) h j = τ(e σ + ρ)h j (2.6) ḡ j = τ(e σ + ρ)g j + [e 2σ τ(e σ + ρ)]l l j + e σ b l j + e σ l b j + b b j (2.7) C jk = τ(e σ + ρ)c jk + (2L) 1 (e σ + ρ)v jk (h j m k ) (2.8) whee τ = L L, m = b - βl 1 l and V jk {} denotes the cyclc ntechange of ndces, j, k and summaton. Fom (2.6) and (2.8) we get the followng, Lemma 2.1 If F n s C-educble Fnsle space then F n s also a C-educble Fnsle space. Fom (2.7), the elaton between contavaant components of the fundamental tenso s gven by ḡ j = (τ(e σ + ρ) 1 g j τ 3 (e σ + e) 1 (e 2σ (1 b 2 ) τ(e σ + e))l l j τ 2 (e σ + ρ) 1 (l b j + l j b ) (2.9)

3 252 MANOJ KUMAR SINGH AND SAMIDHA YADAV EJMAA-2017/5(2) whee b s the magntude of the vecto b = g j b j. Fom (2.8) and (2.9), we get C h j = C h j + (2 L) 1 (h j m h + h h j m + h h m j ) L 1 [ρ + L(2 L) 1 (b 2 β 2 L 2 ))h j + LL 1 m m j ]l n (2.10) Now we shall be concened wth Catan s connecton of F n and F n, ths connecton s denoted by CΓ= (F jk, N k, C jk ). Hee N k =F 0k (= Y j F jk ) and Ch j = g hk C jk. Snce fo a Catan s connecton L j = 0, we obtan Dffeentaton of equaton (2.4b) leads to whee we put ρ k = k ρ = ρ k. If we put k L j = L j N k + L j F k + L F jk. (2.11) k L j = (e σ + ρ) k L j + ρ k L j (2.12) D jk = F jk F jk (2.13) then the dffeence Djk s obvously a tenso of (1.2) type. In vtue of (2.11) equaton (2.12) s wtten n the tensoal fom as, (e σ + ρ)(l j D 0k + L j D k + L D jk = ρ k L j (2.14) In ode to fnd the dffeence tenso Djk, we constuct supplementay equaton to (2.14) fom (2.4a) we obtan Fom L j = 0 equaton (2.15) s wtten n the fom ρ j L = e σ j L + j b (2.15) L N j + L F j = (e σ + ρ)l N j + (L + b )F j + b j By means of (2.4) and (2.13) ths equaton may be wtten n the tensoal fom as, (e σ + ρ)l D 0j + (l + b )D j = b j (2.16) To fnd the dffeence tenso Djk we have the followng[4], Lemma 2.2The system of algebac equaton (1)L A = B (2)(l + b )A = B has a unque soluton A fo gven B and B such that B l = 0, The soluton s gven by A = LB + τ 1 (B LB β )l. whee subscpt β denote the contacton by b Now we gve the followng esult. Theoem 2.1The Catan s connecton of F n s completely detemned by equaton (2.14) and (2.16) n tems of F n. It s obvous that (2.16) s equvalent to the two equatons, (e σ + ρ)(l D 0j + L j D 0) + 2(l + b )D j = 2E j (2.17) (e σ + ρ)(l D 0j L j D 0) = 2F j (2.18) Whee we put, 2E j = b j + b j, 2F j = b j b j (2.19) on the othe hand (2.14) s equvalent to 2(e σ + ρ)l j D k + (e σ + ρ)(l j D 0k + L jk D 0

4 EJMAA-2017/5(2) ON H-TRANSFORMATION OF SOME SPECIAL FINSLER 253 contactng (2.17) wth y j, we get Smlaly fom (2.18) and (2.20), we obtan contactng of (2.21) wth y gves L k D oj) = ρ k L j + ρ L jk ρ j L k (2.20) (e σ + ρ)l D (l + b )D 0 = 2E 0. (2.21) (e σ + ρ)l D 00 = 2F 0 (2.22) (e σ + ρ)(l D 0j + L j D 0 + L j D 00) = ρ 0 L j (2.23) (l + b )D 00 = E 00 (2.24) Now fst consde (2.22) and (2.24) and apply lemma (2.1) to obtan, D 00 = (e σ + ρ) 1 2LF 0 + τ 1 (E 00 2L(e σ + ρ) 1 F β0 )l (2.25) whee we put F 0 = g j F j0 Secondly we add (2.18) and (2.23) to obtan whee we put L D 0j = G j (2.26) G j = (2(e σ + ρ)) 1 (2F j + ρ 0 L j (e σ + ρ)l j D 00). (2.27) The equaton (2.21) s wtten n the fom whee we put Substtutng fom (2.25) n (2.27), we obtan (l + b )D 0j = G J (2.28) G j = E j0 2 1 (e σ + ρ)l j D 00. (2.29) G j = (e σ + ρ) 1 [F j LL j F 0 + L j ((e σ + ρ)e 00 2LF β0 + Lρ 0 )(2L) 1 ] (2.30) By vtue of (2.22),G j ae wtten as G j = E j0 F j0 (2.31) Thus we have obtaned the system of equaton s (2.26) and (2.28),and applyng lemma (2.2) to these equaton s we obtan whee we put G j = g G j Fnally fom (2.20) and (2.17), we get whee we put D 0j = LG j + τ 1 (G j LG βj )l (2.32) L D jk = H jk (l + b )D jk = H jk (2.33) H jk = E jk (eσ + ρ) (L j D0k + L k D 2 0j) H jk = (2(e σ + ρ)) 1 (ρ k L j + e j L k ρ L kj ) 1 2 (L jd 0k + L k D 0j L kj D 0) Now applyng lemma (2.1) to equaton (2.33), we get D jk = LH jk + τ 1 (H jk LH βjk )l (2.34)

5 254 MANOJ KUMAR SINGH AND SAMIDHA YADAV EJMAA-2017/5(2) whee we put H jk = gh H hjk. By vtue of (2.32) H jk and H jk ae wtten n tems of known quanttes, H jk = 1 2 L(L kjg L j G k L k G j) + L j A k + L k A j L jk A (2.35) H jk = E jk (e σ + ρ) L 2 (L jg k + L k G j) (2.36) whee A = (2(e σ + ρ)) 1 ρ + (2τ) 1 (G LG β ) 3. The h-toson tenso R hjk of F n Let F n be a locally Mnkowsk space whose fundamental functon L s expessed by L(y) = (g j y y j ) 1 2 (y = dx ) n tems of an adoptable co-odnate system x. The connecton paamete CΓ of the cetan connecton of F n s gven by F jk = 0, N j = F 0j = 0, C jk = g C jk (3.1) Thus the h-covaant dffeentaton X j of a covaant vecto feld X may be wtten as X j = j X. In vew of (2.13), (2.32) and (3.1), the connecton paamete N j of F n may be wtten as The value of G j n (2.30) may be wtten as whee and N j = LG j + τ 1 (G j LG βj )l (3.2) G j = (e σ + e) 1 {A j + L 1 (F j0 (l + F 0 l j ) + L j ) + Gh j } (3.3) G = (2L L) 1 ((e σ + ρ)e 00 2LF β0 + Lρ 0 ) (3.4) The h-toson tenso R hjk of (M n, L s defned A j = F j 2C j F 0 (3.5) R hjk = V (j,k) { h h ( k N j N k N j )} (3.6) The symbol V (j,k) denotes the ntechange of (j, k) and substacton. In vew of (2.6), we have R hjk = V (j,k) {(e σ + ρ) LL h ( k N j N j N j )} (3.7) By vtue of (3.1) and (2.13) equaton (2.26) may be wtten as L h N j = G h, by whch we wte L h h N j = G h j and V (j,k) {L h N k N j } = V (j,k) (LG k G hj ) Thus (3.7) may be wtten as R hjk = (e σ + ρ)v (j,k) { L(G hj k LG k G hj )} (3.8) By vtue of equaton (3.3), we have G kj h = (e σ + ρ) 1 [A hj k + L 1 (l h F j0 k + l j F h0 k ) + G kh hj ] (e σ + ρ) 2 ρ k (A hj + L(l h F j0 + l j F h0 ) + G h h j ) (3.9) G hj = (e σ + ρ) 1 [ 2(F m0 C m hj + C m hjf m ) + Gh hj + (G ρ 0 (2L) 1 )(2C hj L 1 (l h h j + l j h h )) + L 2 (h h l h l )F j0

6 EJMAA-2017/5(2) ON H-TRANSFORMATION OF SOME SPECIAL FINSLER (h j l j l )F h0 ) + L 1 ((l h F j + l j F h (ρ j h h ρ h h j )). (3.10) Fom equaton (3.3) and (3.10), we get (e σ + ρ) 2 V (j,k) {G k G hj } = V (j,k) { [A j G + G j G + L 1 l j (F0 G + G 2 ) L 2 G(F j0 2 1 ρ 0 l j Lρ j )]h hk + 2A j(f s0 C s hk + C s hkf s + (2L) 1 ρ 0 C hk ) + 2GF s0 ( j Chk s + 2CjC s hk) L 2 (A hj F k0 F h0 F jk F0 F j l h l k ) L 1 [A jf h l k +2F0 ((F s0 Chj s +Chj(F s s )l k +2F0 CjF s sk l h )] L 2 ρ 0 C hj Fo l k +2 1 L 2 ρ 0 (l h A jk + l j A hk + L 1 l h l j A k0 ) L 1 (ρ j A hk ρ h A jk ) L 2 ρ j (l h A k0 + l h F ho )}. (3.11) on substtutng (3.9) and (3.11) n (3.8) and we get Theoem 3.1The h-toson tenso R hjk of the Fnsle space F n s wtten n the fom R hjk = (e σ + ρ) 1 V (j,k) { LL G j h hk + L 2 K hjk + (l h k jk + l j K kh ) l h l j k 0k } (3.12) whee G j = A j G + G j G L 1 (G j (e σ + ho) (F L 2 GF j L 2 G(Lρ j ρ 0 l j ). G + G 2 )l j ) K jk = K jok τ(a kf j 2GC s jkf s0 + L 1 (2F j0 F k0 + ρ 0 A jk +ρ 0 C jk F 0 + (2L) 1 (ρ k F j0 + ρ j F k0 )) K hjk = τ[l 1 (e σ +ho)a hjk 2A j(f s0 C s hk +C s hkf s ) 2GF s0 ( j C t hk +2C s jc hk) +L 2 (A hj F k0 F h0 F jk ) + ρ 0 L 1 C hj A h + (2L) 1 (ρa hk + ρ h A jk )]. If the Fnsle space F n s of scala cuvatue R then we have the equaton R 0j = R L 2 h j [4]. If the scala R s constant then F n s sad to be of constant cuvatue. Fom equaton (3.12) the contacted h-toson tenso R 0j of F n s gven by R 0j = (e σ + ρ) 1 ( LLG 0h j + L 2 W j L(l W j0 + l j W 0 ) + W 00 l l j ) (3.13) whee we put W j = K 0j K j0 + K j and W j s symmetc n the ndces and j. Equaton R 0j = R L 2 h j may be wtten as R 0j = τ(e σ + ρ) R L 2 h j.thus fom equaton (3.13) we get the followng : Theoem 3.2 Let F n be a Fnsle space wth the metc L = e σ L + β whee L = (g j (y)y y j ) 1/2, β = b (x, y)y and b s an h vecto n (M n, L). If F n s of scala cuvatue R then the matx [λh j W j ] s of ank less than thee whee λ = τ((e σ + ρ) 2 τ 2 R G 0 ). Now we consde the case F j = 0. In ths case A j = 0, K jk = 0, K j = 0 and hence W j = 0 holds good. Theefoe the tenso R 0j of F n s educed to the fom R 0j = (e σ + ρ) 1 LLG 0 h j. Consequently we have the followng Theoem 3.3Let F n be an above Fnsle space. If the condton F j = 0 s satsfed, then F n s of scale cuvatue R = ((e σ + ρ)τ) 2 G 0. Now we get the followng, Theoem 3.4 In the above theoem f the scala R s constant, then R = 0 and the space F n s a locally Mnkowskan space. Poof. Fom equaton (2.3) and F j = 0 we get 2 F j = L 1 (ρ j h ρ h j ) = 0 (3.14)

7 256 MANOJ KUMAR SINGH AND SAMIDHA YADAV EJMAA-2017/5(2) whch afte contacton wth y gves ρ 0 = 0. Thus contactng equaton (3.14) wthg j we get ρ j = 0.Theefoe the scala R s wtten n the fom R = ((e σ + ρ)τ) 2 (G 2 b L 1 (e σ + ρ)g 0 (3.15) Fom equaton (3.15) and G = (e σ + ρ)e 00 (2L L) 4 t follows that the condton R = constant s wtten n the fom [2βE E (L 4 + 6L 2 β 2 + β 4 )C] + 2L[E β(L 2 + β 2 )C] = 0 (3.16) fom above equaton we see that fst backet s a fouth degee polynomal and second backet s thd degee polynomal n y.theefoe we wte 2βE E (L 4 + 6L 2 β 2 + β 4 )C = 0 (3.17) E β(L 2 + β 2 )C = 0 (3.18) Fom equaton (3.17) and (3.18) we get 3E 2 00 = 4C(L 2 β)(l 2 + 3β 2 ) (3.19) If C 0 then n vew of F j = 0 and b 0 0 = 0, the h-covaant devatve of (3.19) gves 3E 00 0 = 8Cβ(L 2 3β 2 ) (3.20) Elmnaton of E 00 0 fom (3.18) and (3.20) gves L 2 βc = 0 fom whch we get β = 0 as L 2 C 0. Snce j β = b j, theefoe b = 0 gves E j = 0. Hence Equaton (3.19) gves C = 0. Ths contadcts ou assumpton C o. Hence the scala R = C = 0 and fom equaton (3.19) we get E 00 = 0. Snce F j = 0 gves ρ = 0, theefoe E 00 = 0 mples F j = 0 that s b j = j b = 0. Thus b does not contan x. Hence F n s Locally Mnkowskan space. 4. The hv-toson tenso P jk of F n The hv- toson tenso P hjk of F n s defned as P hjk = C hjk 0 = y Chjk Chjk N 0 V (hjk) { C hj F ko } (4.1) whee V (jk) denotes the cyclc ntechange of ndces jk and summaton. In vew of (2.8) and P hjk =C hjk 0 =0, we obtan y Chjk = C hjk 0 = 2( LG + F β 0 )C hjk + V (hjk) {(2L) 1 (ρ 0 m k + (e σ + ρ)(b k 0 Lτ 1 G o l k )h hj } (4.2) Chjk = τ(e σ + ρ) C hjk + L 1 (e σ + ρ)c hjk m + V (hjk) {(e σ + ρ)l 1 C hj m k + (2L 2 ) 1 (e σ + ρ)(n k + (ρ βl 1 )h k ) + 2L 2 ) 1 (e σ + ρ)h h n jk } (4.3) whee we put n j = l m j + l j m, theefoe fom (3.2), (3.3) and (4.3), we get Chjk N 0 = 2 L C hjk F0 (2 LG Lρ 0 2F β 0 )C hjk + V (hjk) {2F 0 Chjm k L 1 F h0 n jk h hj (L 1 F β0 l k L 1 )(ρ βl 1 )F k0 + (G (2L) 1 ρ 0 )m k )} (4.4) By vtue of equaton (3.2), (3.3) and (2.8), we have V (hjk) { C hj F k0 } = 3 LGC hjk + V (hjk) { LChj(A k + L 1 F 0 l k ) 2ChjF 0 m k + L 1 F h0 n jk h j(a βk + L 1 F β0 l k + L 2 βf k0 + 3Gm k )} (4.5)

8 EJMAA-2017/5(2) ON H-TRANSFORMATION OF SOME SPECIAL FINSLER 257 fom equaton (4.2), (4.4) and (4.5) equaton (4.1) gves the followng Theoem 4.1 The hv-toson tenso P hjk of a Fnsle space F n s wtten as P hjk = 2τT hjk F 0 + ( LG τρ 0 )C hjk + V (hjk) {τc hj(f 0 l k + LF k ) + h hj P k }. whee 2P k = A βk + L 1 [(e σ + ρ)e k0 + (τ ρ)f k0 (F β0 + 2 LG τρ 0 )l k G mk T hjk = LC hjk + C hj l V (hjk) {C jk l h } If the condton F j = 0 s satsfed then the hv-toson tenso P hjk of F n s gven by P hjk = ( LG τρ 0 )C hjk + V (hjk) {h hj P k } (4.6) whee G = (2L L) 1 {(e σ + ρ)e 00 + Lρ 0 } 2P k = L 1 [(e σ + ρ)f k0 (2 LG τρ 0 )l k ] Gm k. Now we shall teat a Landsbeg space of F n. Such a space s by defnton, a Fnsle space wth vanshng of hv-toson tenso P hjk. On the othe hand a Fnsle space F n wth C hj k = 0 s called a Bewald space. Theoem 4.2 Let F n (n 3) be a Fnsle space wth the metc L = e σ L + β whee L = (g j (y)y y j ) 1/2, β = b (x, y)y and b s an h vecto n (M n, L). In the case F j = 0 f F n s a Landsbeg space then F n s a Bewald space. Poof. The condton ( LG τρ 0 ) = 0 mples that E 00 = 0.e. E j = 0 and hence F j = 0 t follows that b s ndependent of x. Thus F n s locally Mnkowskan. In the case ( LG τρ 0 ) 0,fom equaton (4.6) t follows that P hjk s equvalent to C hjk = ( LG τρ 0 ) 1 V (hjk) {h hj P k } Hence F n s C-educble, then F n s also C-educble. Then F n s a Bewald space [5]. 5. Some cuvatue Popetes of Fnsle space F n The v-cuvatue tenso S jkl of F n wth espect to Catan connecton CΓ s wtten n the fom S jkl = C lm C m jk C km C m jl (5.1) A Fnsle space F n (n 4) s called S3 -lke[6] f the v-cuvatue tenso S jkl s of the fom S S jkl = (n 1)(n 2) (h kh jl h l h jk ) (5.2) whee the scala S s functon of co-odnates only. A non-remannan Fnsle space F n (n 5) s called S4-lke [7] f S jkl s of the fom of L 2 S jkl = h k M jl + h jl M k h l M jk h jk M l (5.3) wheem j s symmetc and ndcatoy tenso.

9 258 MANOJ KUMAR SINGH AND SAMIDHA YADAV EJMAA-2017/5(2) A non-remannan Fnsle space F n of dmenson n 3 s called C-educble [5] f the (h)hv-toson tensoc jk s wtten n the fom of C jk = 1 n + 1 (C h jk + C j h k + C k h j ) (5.4) whee C = C j j A Fnsle space F n (n 2) wth C 2 = g j C C j 0 s called C2-lke[8], f (h)hvtoson tenso C jk s wtten n the fom of C jk = 1 C 2 C C j C k (5.5) A Fnsle space F n (n 3) wth C s called sem C-educble[7], f the (h)hv-toson tenso C jk s of the fom of C jk = P (n + 1) (h jc k + h jk C + h k C j ) + q C 2 C C j C k (5.6) whee p and q = (1 p) does not vansh p s called the chaactestc scala of the F n. Fom equaton (5.1) and (2.8) and (2.10), the v-cuvatue tenso of F n s gven by S jkl = τ(e σ + ρ)s jkl + h jk d l + h l d jk h jl d k h k d jl (5.7) ρ whee d l = (e σ + ρ){ m2 8L L h l + 2L h 2 l LL m m l } and m 2 = m m, m = g j m j Let us suppose that F n be an S3-lke Fnsle space, then fom equaton (5.2), (2.6) and (5.7), we have S jkl = h l P jk + h jk P l h jl P k h k P jl whee P j = {τ(e σ + ρ)} 1 S (n 1)(n 2) h j Whch shows that F n s an S4 - lke Fnsle space. Let us suppose that F n s an S4-lke Fnsle space then fom the equaton (5.3), (2.6) and (5.7), we obtan S jkl = h jk B l + h l B jk h jl B k h k B jl whee B j = {(τ(e σ + ρ)) 1 d j L 2 M j } Whch show that F n s an S4 - lke Fnsle space. Next let us suppose that S jkl = 0 then fom equaton (2.6) and (5.7),we get S jkl = h jk E l + h l E jk h jl E k h k E jl Whee E jk = {τ(e σ + ρ)} 1 d jk whch shows that F n s an S4-lke Fnsle space Summazng all these esults we get the followng. Theoem 5.1 If F n s any one of the followng Fnsle spaces (a) S3-lke Fnsle space (b) S4-lke Fnsle space (c) A Fnsle space wth vanshng v-cuvatue tenso,

10 EJMAA-2017/5(2) ON H-TRANSFORMATION OF SOME SPECIAL FINSLER 259 then F n s an S4 - lke Fnsle space. The v-cuvatue tenso S jkl of a C-edusble Fnsle space has been obtaned by Matsumoto[5] s of the followng fom S jkl = (n + 1) 2 (h l C jk C l h k c jl h jl C k ) (5.8) whee (a) C j = 2 1 C 2 h j + C C j Snce C j s a symmetc and ndcatoy tenso theefoe (5.8) shows that F n s an S4-lke Fnsle space. Thus n vew of theoem (5.1) we have the followng esult. Theoem 5.2 If F n s a C-educble Fnsle space, then F n s an S4-lke Fnsle space. Fom equaton (5.1) and (5.5) t can be shown that the v-cuvatue tenso S jkl of a C2-lke Fnsle space vanshes. Theefoe n vew of theoem (5.1) we have the followng. Theoem 5.3 If F n s a C2-lke Fnsle space, then F n s an S4-lke Fnsle space. Acknowledgement: The authos ae thankful to efeee fo hs/he valuable comments and obsevatons whch help n mpovng the pape. Refeences [1] M.Matsumoto: On Fnsle space wth Randes metc and specal foms of mpotant tensos, J. Math. Kyoto Unv., 14 (1974), [2] H.Izum: Confomal tansfomaton of Fnsle spaces. II., Tenso. N.S., 34 (1980), [3] G.Randes: On the assymetcal metc n the fou-space of geneal elatvty., Phy. Rev., 56(2) (1941), [4] M.Matsumoto: Foundatons of Fnsle geomety and specal Fnsle spaces., Kasesha Pess Sakawa,Otsu, Japan, (1986). [5] M.Matsumoto: On C-educble Fnsle spaces, Tenso. N. S. 24 (1972), [6] M.Matsumoto: Theoy of Fnsle spaces wth (α, β)-metc, Rep. Math.Phy. 31 (1992), [7] M.Matsumoto and C.Shbata: J. Math. Kyoto Unv., 19 (1979), [8] M.Matsumoto and S.Numata: Tenso(N.S), 34 (1980), [9] B.N.Pasad: On the toson tensos R jk and P jk of Fnsle spaces wth a metc ds = (g j dx dx j ) 1/2 + b (x, y)dx Indan J. Pue. Appl. Math., 21(1) (1990), [10] A.Taleshan,D.M.Saghal and S.A.Aab: Confomal h-vecto-change n Fnsle spaces, Jounal of Mathematcs and Compute Scence 7 (2013) [11] S.H.Abed: Confomal β-changes n Fnsle spaces, Poc. Math. Phys. Soc. Egypt, 86 (2008) Manoj Kuma Sngh Depatment of Mathematcs, UIET, CSJM Unvesty,Kanpu, (U.P.) Inda E-mal addess: ms84ddu@gmal.com Samdha yadav Depatment of Mathematcs, UIET, CSJM Unvesty,Kanpu, (U.P.) Inda E-mal addess: samdha.yadav14@gmall.com