New Cartan s Tensors and Pseudotensors in a Generalized Finsler Space

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1 Flomat 8: 04, 07 7 DOI 0.98/FIL4007C Publhed by Faculty of Scence and Mathematc, Unverty of Nš, Serba valable at: htt:// New Cartan Tenor and Peudotenor n a Generalzed Fnler Sace Mlca D. Cvetkovć a, Mlan Lj. Zlatanovć b a Unverty of Nš, Faculty of Scence and Mathematc, 8000 Nš, Serba, and College for led Techncal Scence, 8000 Nš, Serba, b Unverty of Nš, Faculty of Scence and Mathematc, 8000 Nš, Serba btract. In th work we defned a generalzed Fnler ace GF N a N-dmenonal dfferentable manfold wth a non-ymmetrc bac tenor j x, ẋ, whch ale that j θmx, ẋ = 0, θ =,. Baed on non-ymmetry of bac tenor, we obtaned ten Rcc tye dentte, comarng to two knd of covarant dervatve of a tenor n Rund ene. There aear two new curvature tenor and ffteen magntude, we called curvature eudotenor.. Introducton Fnler geometry a natural and fundamental generalzaton of Remann geometry. It wa frt uggeted by Remann a early a 854 [5], and tuded ytematcally by Fnler n 98 [5]. The name Fnler Geometry wa frt gven by J. Taylor n 97. The non-ymmetrc connecton wa nteretng to many author:. Yano [],.C. Shamhoke [7], S.Mnčć [8]-[0], S. Manoff [6], C.. Mhra[] and many other: [7], [0], []. Fnler [5] ace F N are N-dmenonal manfold the nfntemal dtance between two neghborng ont x, x + dx gven by: d = Fx, dx, =,..., N, F requred to atfy ome roerte [6]: Fx, dx > 0; Fx, λdx = λfx, dx, for any λ > 0; 3 The quadrc form F x, dx dx dx j ξ ξ j > 0, for all vector ξ and any x, dx. Then, the metrc tenor defned from a: j = F x, ẋ, ẋ ẋ j 3 00 Mathematc Subject Clafcaton. Prmary 5345; Secondary 53B05, 53B40 eyword. The generalzed Fnler ace, Rcc tye dentte, h-dfferentaton. Receved: 6 May 03; cceted: Setember 03 Communcated by Ljubca Velmrovć Reearch uorted by the reearch roject 740 of the Serban Mntry of Scence Emal addree: mlcacvetkovc@bb.r Mlca D. Cvetkovć, zlatmlan@mf.n.ac.r Mlan Lj. Zlatanovć

2 M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, ẋ = dx dt are the tangent vector to a curve C : x = x t n the manfold ace, or element of the tangent ace T n x at ont x. Ung the econd condton n, j are homogeneou of degree zero n the other et of varable and we can wrte: d = j x k, dx k dx dx j. 4 Defnton.. The generalzed Fnler ace GF N a dfferentable manfold wth non-ymmetrc bac tenor j x, ẋ, j x, ẋ j x, ẋ, = det j 0. 5 Baed on 5, t can be defned the ymmetrc, reectvely, antymmetrc art of j : j = j + j, j = j j, 6, followng [7], t true that: a j = F x, ẋ j, b ẋ ẋ j ẋ = 0, 7 k F a metrc functon n GF N, havng the roerte known from the theory of uual Fnler ace. In the aer [8 0, 3, 4] we tuded generalzed Fnler ace. Introducng a Cartan tenor C jk, mlar a n F N, we have: C jk x, ẋ de = f j,ẋ = k 7b j,ẋ = k 4 F, 8 ẋ ẋ j ẋ k = 7b gnfe equal baed on 7b. We can conclude that C jk ymmetrc n relaton to each ar of ndce. lo, we have: C jk de f = h C jk = 8 h C jk = h C jk = C kj. 9 In GF N the next equaton are vald: C jk ẋ = C jk ẋ j = C jk ẋ k = 0. 0 One obtan coeffcent of non-ymmetrc affne connecton n the Cartan ene []: Γ jk = γ jk hl C jl Γ k + C klγ j C jkγ l ẋ Γ kj, Γ.jk = Γ r jk r = γ.jk C j Γ k + C kγ j C jkγ ẋ Γ.kj. We defned the coeffcent: Γ jk = γ jk hl C kl Γ j + C jlγ k C kjγ l ẋ Γ kj, 3 Γ.jk = Γ r jk r = γ.jk C k Γ j + C jγ k C kjγ ẋ Γ.kj. 4 Let u denote: T jk x, ẋ = Γ jk Γ kj, T jk x, ẋ = Γ jk Γ kj, Γ jk = Γ jk + Γ kj, Γ jk = Γ jk + Γ kj,

3 M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, a toron tenor of the connecton Γ, Γ, reectvely. Baed on non-ymmetry of the coeffcent of the connecton, t can be defned two knd of h-covarant dervatve: T j m = T j,m + T j Γ m TΓ jm T j,ṡ Γrmẋr, T j m = T j,m + T j Γ m T Γ mj T j,ṡ Γmrẋr. By the rocedure that mlar n a Fnler ace, t can be roved that covarant dervatve 5 of a tenor alo a tenor. Theorem.. For the tenor j x, ẋ baed on both knd of dervatve 5 n GF N t vald: 5 j θmx, ẋ = 0, θ =,. 6 Proof. Startng from 5, we get: j mx, ẋ = j,m j,ṗ Γ rmẋ r Γ m j Γ jm = = j,m C j Γ rmẋ r Γ.jm + 8, Γ j.m = 0. 7 The ame reult one obtan for j m : j mx, ẋ = j,m j,ṗ Γ mrẋ r Γ m j Γ mj = = j,m C j Γ mrẋ r Γ j.m + Γ.mj = 0, 8 8, and we have roved 6.. Rcc tye dentte for h-dfferentaton n GF N It known that n Fnler ace there only one Rcc dentty for h-dfferentaton, correondng to alternated covarant dervatve of the nd order. In the cae of non-ymmetrc affne connecton there are 0 oblte to form the dfference: a r r u t t v m n ar r u t t v m λ µ ν ω n λ, µ, ν, ω =,, 9, denote two knd of covarant dervatve baed on.7,.8, and we can obtan ten Rcc tye dentte and two tenor of curvature. Correondng dentte n GF N may be roved by total nducton method. The mentoned oblte are obtaned for thee combnaton: λ, µ; ν, ω {, ;,,, ;,,, ;,,, ;,,, ;,,, ;,,, ;,,, ;,,, ;,,, ;, }. 0 For fndng the general cae baed on., we frtly oberve the cae of a tenor a x, ξ. Let u obtan the cae when the vector feld ξ l tatonary comarng to the frt knd of covarant dervatve,.e. ξ l h x, ξ = 0, ξl h x, ξ 0.

4 M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, nd we have: ξ l,h = ξl x h = Γ l rh ξr = Gl x, ξ ẋ h = G l,ḣ. Then, we obtaned, for examle, a j mn =a j m,n + a j m,ṡξ ș n + Γ na j m Γ jn a m Γ mna j = =a j,mn + a j,nṡ ξș m + a j,ṡ ξș mn + Γ m,na j + Γ ma j,n Γ jm,n a Γ jm a,n+ + a j,mṡ ξș n + a j, lṡ ξl,mξ ș n + a j, l ξl,mṡ ξș n + Γ m,ṡ a j ξș n + Γ ma j,ṡ ξș n Γ jm,ṡ a ξ ș n Γ jm a,ṡ ξș n+ + Γ na j,m + Γ na j,ṡ ξș m + Γ nγ ma j Γ nγ jm a Γ jn a,m Γ jn a,ṡ ξș m Γ jn Γ ma + Γ jn Γ ma Γ mna j. 3 nd mlar: a j m n = a j,mn + a j,nṡ ξș m + a j,ṡ ξș mn + Γ m,na j + Γ ma j,n Γ jm,n a Γ jm a,n+ + a j,mṡ ξș n + a j, lṡ ξl,mξ ș n + a j, l ξl,mṡ ξș n + Γ m,ṡ a j ξș n + Γ ma j,ṡ ξș n Γ jm,ṡ a ξ ș n Γ jm a,ṡ ξș n+ + Γ na j,m + Γ na j,ṡ ξș m + Γ nγ ma j Γ nγ jm a Γ nj a,m Γ nj a,ṡ ξș m Γ nj Γ ma + Γ nj Γ ma Γ nma j. 4 a j m n = a j,mn + a j,nṡ ξș m + a j,ṡ ξș mn + Γ m,na j + Γ ma j,n Γ mj,n a Γ mj a,n+ + a j,mṡ ξș n + a j, lṡ ξl,mξ ș n + a j, l ξl,mṡ ξș n + Γ m,ṡ a j ξș n + Γ ma j,ṡ ξș n Γ mj,ṡ a ξ ș n Γ mj a,ṡ ξș n+ + Γ na j,m + Γ na j,ṡ ξș m + Γ n Γ ma j Γ n Γ mj a Γ jn a,m Γ jn a,ṡ ξș m Γ jn Γ ma + Γ jn Γ ma Γ mna j. 5 a j mn = a j,mn + a j,nṡ ξș m + a j,ṡ ξș mn + Γ m,na j + Γ ma j,n Γ mj,n a Γ mj a,n+ + a j,mṡ ξș n + a j, lṡ ξl,mξ ș n + a j, l ξl,mṡ ξș n + Γ m,ṡ a j ξș n + Γ ma j,ṡ ξș n Γ mj,ṡ a ξ ș n Γ mj a,ṡ ξș n+ + Γ na j,m + Γ na j,ṡ ξș m + Γ n Γ ma j Γ n Γ mj a Γ nj a,m Γ nj a,ṡ ξș m Γ nj Γ ma + Γ nj Γ ma Γ nma j. 6 lo, we have a j,ṡ ξș mn + a j, l ξl,mṡ ξș n = a G j,ṡ + a G l G x n ẋ m j, l ẋ ẋ m ẋ n = Γ = a rm j,ṡ x n ẋr Γ lm + Γ rm ẋ r G l ẋ l,ṅ = = a j,ṡ Γ rm,n Γ lm Γ l rn Γ l ẋr rm, l Gl,ṅ. 7

5 M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, We condered the dfference: a j mn a j nm = a j,mn + a j,nṡ ξș m + a j,ṡ ξș mn + Γ m,na j + Γ ma j,n Γ jm,n a Γ jm a,n+ + a j,mṡ ξș n + a j, lṡ ξl,mξ ș n + a j, l ξl,mṡ ξș n + Γ m,ṡ a j ξș n + Γ ma j,ṡ ξș n Γ jm,ṡ a ξ ș n Γ jm a,ṡ ξș n+ + Γ na j,m + Γ na j,ṡ ξș m + Γ nγ ma j Γ nγ jm a Γ jn a,m Γ jn a,ṡ ξș m Γ jn Γ ma + Γ jn Γ ma Γ mna j a j,nm a j,mṡ ξș n a j,ṡ ξș nm Γ n,ma j Γ na j,m + Γ jn,m a + Γ jn a,m a j,nṡ ξș m a j, lṡ ξl,nξ ș m a j, l ξl,nṡ ξș m Γ n,ṡ a j ξș m Γ na j,ṡ ξș m + Γ jn,ṡ a ξ ș m + Γ jn a,ṡ ξș m Γ ma j,n Γ ma j,ṡ ξș n Γ mγ na j + Γ mγ jn a + + Γ jm a,n + Γ jm a,ṡ ξș n + Γ jm Γ na Γ jm Γ na + Γ = mn a j jmn a a j,ṡ rmn ẋ r T mna j, nma j = 8 mn = Γ m,n Γ n,m + Γ mγ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ. 9 Theorem.. In the generalzed Fnler ace GF N, for h-dfferentaton, the frt Rcc tye dentty exreed: mn a nm = r αmn, ṡ rmn ẋ r Tmna, 30 α= β= gven by 9 and = a r + r u t t v, = a r r u t t β r β+ t v. 3 Ung.6 t eay to rove that: mn ẋ r = We condered: G x n ẋ m G x m ẋ n + G G m ẋ n G G n ẋ m = G,nṁ G,ṅm + G mg ș ṅ G ng ș ṁ, 3 G mn = a j mn a j nm = mn a j jmn a a j,ṡ rmn ẋ r T and fnally, we get: = mn + rmn l r a j a j mn a j nm = R mn a j R jmn a j a j rmn l r T G ẋ n ẋ m = G nm. mna j = jmn + j rmn l r a a j rmn l r T mna j mna j, 33, 34 we denoted the thrd tenor of Cartan, our the frt new curvature tenor ee Rund: R mn = mn + C rmn ẋ r = = Γ m,n Γ n,m + Γ mγ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ + C G ș nṁ G ș ṅm + G qmg q,ṅ 35 G qng q,ṁ,

6 M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, 07 7 and alo t vald: R mn ẋ = mn ẋ. 36. For the next dfference, we got: a j mn a j nm = mn a j jmn a + a j,ṡ rmn ẋ r + T = mn + rmn l r a j = R mn a j R jmn a j a j rmn l r + T mna j = jmn + j rmn l r a a j rmn l r + T mna j, mna j = 37 jmn = Γ mj,n Γ nj,m + Γ mj Γ n Γ nj Γ m Γ mj,ṡ Gș ṅ + Γ nj,ṡ Gș ṁ, 38 and the econd curvature tenor gven wth: R mn = mn + C rmn ẋ r = = Γ mj,n Γ nj,m + Γ mj Γ n Γ nj Γ m Γ mj,ṡ Gș ṅ + Γ nj,ṡ Gș ṁ + C G ș nṁ G ș ṅm + G qmg q,ṅ 39 G qng q,ṁ. It eay to rove that: R mn ẋ = mn ẋ. 40 Theorem.. In the generalzed Fnler ace GF N, for h-dfferentaton, the econd Rcc tye dentty exreed: mn a nm = r αmn, ṡ rmn ẋ r + T mna, 4 gven by 38. α= β= 3. For the dfference, we can get: a j m n a j n m = mn a j jmn a + a j<mn> + a j mn a,ṡ rmn ẋ r + T = mn + rmn l r a j + a j<mn> + a j mn a,ṡ rmn ẋ r + T = B mn a j B mna j = jmn + j rmn l r a a j rmn l r + mna j = jmn a j a j rmn l r + a j<mn> + a j mn a,ṡ rmn ẋ r + T mna j, 4 mn = Γ m,n Γ n,m + Γ m Γ n Γ n Γ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ, 43 mn = Γ m,n Γ n,m + Γ mγ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ, 44 B mn = mn + C rmn ẋ r, B mn = mn + C rmn ẋ r, 45 a j<mn> = M ma j,n + a j,ṡ ξș n M jm a,n + a,ṡ ξș n, 46

7 M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, a j mn = Γ mγ jn Γ m Γ nj a. It eay to rove that: 47 B mn ẋ = mn ẋ, B mn ẋ = mn ẋ. 48 Theorem.3. In GF N, for h-dfferentaton, the 3 rd Rcc tye dentty exreed by: m n a n m = r αmn, ṡ rmn ẋ r + α= β= + <mn> + mn + T mn and are gven by equaton 43, 44 and <mn> = α= M m,n +,ṡ ξș n β=, 49 M t β m,n +,ṡ ξș n, 50 u a r r u t t v mn = α= β= α<β v + α= β= α<β Γ [m Γ [t α m Γ r β n ] Γ tα nt β ] r β, α= β= Γ Γ [m nt β ] + 5 = a r + r u t t β r β+ t v. 4. We alo condered the dfference: a j m n a j n m = mn a 3 j 4 jmn a a j<mn> a j mn + a,ṡ rmn ẋ r T mna j, 5 3 mn = Γ m,n Γ n,m + Γ mγ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ, 53 4 mn = Γ m,n Γ n,m + Γ m Γ n Γ n Γ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ. 54 Theorem.4. lyng the two knd of covarant dervatve n an nvere order than uual, we obtaned the 4 th Rcc tye dentty n GF N for h-dfferentaton: m n a n m = r αmn 3 4, ṡ rmn ẋ r α= β= 55 <mn> mn T mn, 3 and 4 are gven by 53, 54.

8 5. We condered the dfference: M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, a j mn a j nm = mn a 5 j 6 jmn a + a j<mn> + a j mn a,ṡ rmn ẋ r Γ mna j + Γ mna j, 56 5 mn = Γ m,n Γ n,m + Γ mγ n Γ n Γ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ, 57 6 mn = Γ m,n Γ n,m + Γ m Γ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ. 58 Theorem.5. In GF N, for h-dfferentaton, there the 5 th Rcc tye dentty: mn a nm = r αmn 5 6, ṡ rmn ẋ r + α= β= + <mn> + mn Γ 5 and 6 are gven by 57, 58. u a r r u t t v mn = α= β= α<β v + α= β= α<β Γ [m Γ [t α m Γ r β n ] Γ tα t β n] t β. mn α= β= 59 + Γ mn, Γ Γ [m t β n] r α In the next dfference we got: a j mn a j n m = mn a 7 j 8 jmn a + a j<mn> a j mn + a,ṡ rmn ẋ r M mna j, 6 7 mn = Γ m,n Γ n,m + Γ mγ n Γ n Γ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ, 6 8 mn = Γ m,n Γ n,m + Γ mγ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ. 63 Theorem.6. In GF N for h-dfferentaton, there the 6 th Rcc tye dentty: r αmn 7 mn a n m = α= β= 8 7 and 8 are gven by equaton 6, 63 and u a r r u t t v mn = α= β= α<β α= β= Γ [m] Γ r β n + Γ n Γ r β [m] r β Γ [m] Γ t β n + Γ n Γ [t β n], ṡ rmn ẋ r + <mn> + mn, v + α= β= α<β Γ [t α m] Γ t β n + Γ t α n Γ [t β n] tα.

9 7. Then, we condered the dfference: a j mn a j n m = 9 M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, mn a j 0 jmn a + a j<nm> a j nm + a,ṡ rmn ẋ r Γ mna j + Γ nma j, 66 9 mn = Γ m,n Γ n,m + Γ mγ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ, 67 0 mn = Γ m,n Γ n,m + Γ m Γ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ. 68 Theorem.7. In GF N, for h-dfferentaton, the 7 th Rcc tye dentty vald: mn a n m = r αmn 9 0, ṡ rmn ẋ r + α= β= + <nm> + nm Γ 9 and 0 are gven by equaton 67, 68. mn Γ nm, nd, we condered the dfference: a j mn a j n m = mn a j jmn a a j<nm> + a j nm + a,ṡ rmn ẋ r + Γ mna j Γ nma j, 70 mn = Γ m,n Γ n,m + Γ m Γ n Γ n Γ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ, 7 mn = Γ m,n Γ n,m + Γ mγ n Γ n Γ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ. 7 Theorem.8. In GF N, for h-dfferentaton, there the 8 th Rcc tye dentty: mn a n m = r αmn, ṡ rmn ẋ r α= β= <nm> + mn + Γ mn and are gven by equaton 7, 7 and u a r r u t t v mn = α= β= α<β α= β= Γ mγ r β [n] + Γ [n] Γ r β m 9. In the next dfference, we got: a j mn a j n Γ Γ mγ [nt β ] + Γ [n] Γ mt β nm r β, m = mn a 3 j 4 jmn a a j<mn> + a j nm + a,ṡ v + α= β= α<β rmn ẋ r + M Γ mt α Γ [nt β ] + Γ [nt α ] Γ mt β nma j tα., 75 3 mn = Γ m,n Γ n,m + Γ m Γ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ, 76 4 mn = Γ m,n Γ n,m + Γ m Γ n Γ n Γ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ. 77

10 M. Cvetkovć, M. Zlatanovć / Flomat 8: 04, Theorem.9. In GF N, for h-dfferentaton, there the 9 th Rcc tye dentty: mn a n m = α= 3 mn β= <mn> + nm + M 3 and 4 are gven by equaton 76, nma j, ṡ rmn ẋ r, t lat, we condered the dfference: a j m n a j n m = mn a 5 j 5 jmn a + a,ṡ rmn ẋ r + Γ nma j Γ nma j, 79 5 mn = Γ m,n Γ n,m + Γ m Γ n Γ nγ m Γ m,ṡ Gș ṅ + Γ n,ṡ Gș ṁ. 80 Theorem.0. In GF N, for h-dfferentaton, the 0 th Rcc tye dentty vald: m n a n m = α= 5 gven by equaton mn β= 5, ṡ rmn ẋ r Γ nm Γ nm, 8 The dentty.64 can be wrtten n another form: a r r u t t v m n ar r u t t v n m = α= 3 mn β= 3, ṡ rmn ẋ r, 3 jmn = 5 jmn + Γ nmt j = Γ jm,n Γ nj,m + Γ jm Γ n Γ nj Γ m + Γ nmγ j Γ j + +P jm,ṡ ξș n Γ nj,ṡ ξș m Concluon Baed on non-ymmetry of bac tenor n generalzed Fnler ace, ung h-dfferentaton, we defned two knd of covarant dervatve of a tenor n Rund ene and obtaned ten Rcc tye dentte, two new curvature tenor and ffteen magntude, we called eudotenor. art from attemt at a theoretcal unfcaton of gravtaton and electromagnetc henomena n a ngle geometrcal framework, Fnler ace were alo condered ether a formal rooton of new theoretcal tructure and feld equaton, or more drectly concerned wth exlorng oble obervatonal conequence. Fnler geometry doe have many feld of alcaton, bede geometrcal extenon of theore of gravty. lo, comuter algebra can be very helful to gve Fnler exreon from a choen metrc or connecton.

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