RICCI TYPE IDENTITIES FOR BASIC DIFFERENTIATION AND CURVATURE TENSORS IN OTSUKI SPACES 1

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1 wnov Sad J. Math. wvol., No., 00, RICCI TYPE IDENTITIES FOR BASIC DIFFERENTIATION AND CURVATURE TENSORS IN OTSUKI SPACES Svetlav M. Mnčć Abtract. In the Otuk ace ue made of two non-ymmetrc affne connecton: one for contravarant and the other for covarant ndce. In the reent work we tudy the Rcc tye dentte for the bac dfferentaton and curvature tenor n thee ace. AMS Mathematc Subject Clafcaton 000: 5B05 Key word and hrae: Otuk ace, bac dfferentaton, Rcc tye dentty, curvature tenor and eudotenor.. Introducton T. Otuk ha defned and nvetgated [6] the o-called regular general connecton contng of two affne connecton: contravarant Γ and covarant art Γ. Bede, he ntroduced a tenor feld P of the tye, detpj 0, wth the condton [6],. P j,k Γ kp j Γ jk P 0, the comma gnfe uual artal dervatve,.e. P j,k P j / xk. In ace wth th connecton one defne the o-called bac covarant dervatve, for examle Vj;k Vj,k Γ kv j Γ jk V, and non-bac covarant dervatve, for examle and the correondng dfferental k V j P P q j V q;k, 4 DV j V j;kdx k, The relaton equvalent to DV j k V j dx k. 5 k Q j 0, Suorted by Grant 04M0D of RFNS trough Math. Int. SANU. Faculty of Scence, Ćrla Metodja, 8000 Nš, Yugolava

2 74 S. M. Mnčć Q j P j,.e. 6 P Q j P j Q δ j. Aart from T. Otuk the cted ace have been nvetgated alo by A. Moór [], M. Prvanovć [7], [8], Dj. F. Nadj [5] and other.. Rcc tye dentte for bac dfferentaton of the frt and econd knd.0. The Otuk ace O N defned a an N-dmenonal dfferentable manfold on whch, wth reect to local coordnate x,,..., N, gven a tenor feld Pj detp j 0 and the connecton coeffcent Γ jk, Γ jk, whch are non-ymmetrc n general cae and the relaton n force. Snce the connecton coeffcent Γ jk and Γ jk are generally non-ymmetrc wth reect to j, k, one can defne two knd of bac covarant dervatve for a tenor V of the tye u, v: 7 8 V u m V u,m V u m V u,m Γ m Γm β Γ j β m, Γ mj β β, we have ued the degnaton V 9 u, 0 V u j j β j βj v. From here, for the Kronecker ymbol we have δ j m Γ jm Γ jm δ j m Γ mj Γ mj. In order to form the Rcc tye dentte, we can oberve the dfference havng 0 dfferent cae: V u λ m µ n V u ν n ω m,

3 Rcc tye dentte for bac dfferentaton λ, µ; ν, ω {, ;,,, ;,,, ;,,, ;,,, ;,,, ;,,, ;,,, ;,,, ;,,, ;, }, whch we are to tudy... In the cted work, only the frt knd of covarant dervatve were ued and t ha been roved that ee [6], eq. 7.5, or [5] eq V u m n V u n m β R R mn Γ [mn] V u, 6 R jmn Γ jm,n Γ jn,m Γ jm Γ n Γ jn Γ m, and R n the ame manner exreed by Γ, whle [mn] gnfe the antymmetraton wth reect to m, n wthout dvon by, that 7 Γ [mn] Γ mn Γ nm. The dentty 5 we call the frt Rcc tye dentty for bac dfferentaton n O N, whle the thenor R, R are curvature tenor of the t knd n O N, obtaned by Γ, reectvely Γ... In the ame way one rove that n O N n force the econd Rcc tye dentty for bac dfferentaton 8 V u m n V u n m β R R mn Γ [mn] V u, 9 R jmn Γ mj,n Γ nj,m Γ mj Γ n Γ nj Γ m,

4 76 S. M. Mnčć and n the ame manner R by Γ. The quantte R, R are curvature tenor of the econd knd n O N... For the thrd cae, by vrtue of and 4, we have the next theorem: Theorem. In the ace O N the thrd Rcc tye dentty for bac dfferentaton vald: 0 V u m n V u n m mn β V u <[mn]> V u [mn] Γ [mn] V u, jmn Γ jm,n Γ jn,m Γ jm Γ n Γ jn Γ m, jmn Γ jm,n Γ jn,m Γ mj Γ n Γ nj Γ m, V u <mn> Γ [m],n β Γ [j β m],n, 4 u V u mn β <β β v Γ m Γ β n Γ m Γ β n Γ m Γ nj β Γ m Γ j β n β j Γ j m Γ nj β Γ mj Γ j β n β <β. Proof. We hall rove 0 for the tenor Vjkl h, from one can antcate the general formula 0. So, 5 Further, we have V h jkl m V h jkl,m Γ h mv jkl Γ mv h jkl Γ jm V h kl Γ km V h jl Γ lm V h jk. 6 V h h V jkl m n jkl m h V n jkl m,n Γ h nv Γ jkl nv h m jkl m Γ njv h Γ kl nkv h Γ m jl nlv h Γ m jk nmv h m jkl.

5 Rcc tye dentte for bac dfferentaton Subttutng nto 6 by vrtue of 5, one obtan 7 V h jkl m n V h jkl n m h mn V jkl mn V h jkl jmn V h kl kmn V h jl lmn V h jk V h jkl<[mn]> V h jkl [mn] Γ [mn] V h jkl, V h jkl<mn> Γ h [m] V jkl,n Γ [m] V h jkl,n Γ [jm] V h kl,n Γ [km] V h jl,n Γ [lm] V h jk,n, V h jkl mn Γ h m Γ n Γ h m Γ nv jkl Γ h m Γ nj Γ h m Γ jnv kl Γ h m Γ nk Γ h m Γ knv jl Γ h m Γ nl Γ h m Γ lnv jk Γ m Γ nj Γ m Γ jnv h kl Γ m Γ nk Γ m Γ knv h jl Γ m Γ nl Γ m Γ lnv h jk Γ jm Γ nk Γ mj Γ knvl h Γ jm Γ nl Γ mj Γ lnvk h Γ km Γ nl Γ mk Γ lnvj. h We ee that 7 a artcular cae of 0. So, for the tenor Vjkl h the equaton 0 vald. The cae of vector V m n V n m mn V Γ [m] V,n Γ [n] V,m Γ [mn] V, V j m n V n m jmn V Γ [jm] V,n Γ [jn] V,m Γ [mn] V j, are ncluded n 0, whch can be verfed drectly. In the cae of a vector, the exreon 4 zero. Alo, by drect calculaton we obtan that 8 V V j m n j n m A mn V j A jmn V Γ [m] v j,n Γ [n] V j,m Γ [jm] V,n Γ [jn] V,m Γ m Γ nj Γ m Γ jn Γ n Γ mj Γ n Γ jmv Γ [mn] V j, and th obtaned from 0 too.

6 78 S. M. Mnčć In order to rove 0 by nducton method, uoe that 0 vald, and rove that the correondng equaton vald for a tenor W u u j v. Oberve the tenor 9 V u W u u j v U j v u Alyng 0 to th tenor of the tye u, v, we get 0 V u β m n V u n m W U [mn] Γ [mn] A mn W u u j j vj v U j v u W u u j j v j v U j v u W U <[mn]> W U W U, we have taken nto conderaton that for the bac dfferentaton the Lebnz rule vald. Baed on,4,9, we obtan W U <mn> β [u W U mn U β v Γ [m] Γ [j β m] β <β W,nU W U,n W,nU W U,n, Γ m Γ β n Γ m Γ β n Γ m Γ nj β Γ m Γ j β n Γ j m Γ nj β Γ mj Γ j β n β <β On the other hand, baed on 9, we have j β W W W ]. W m m n n nu W W W m n n W mu W n m {W U m n} [mn] mu n W mu W U nu m [mn]. nu m W U m n [mn] U m n n m

7 Rcc tye dentte for bac dfferentaton Alyng the dentty 8 to the tenor U jv u at the econd bracket, calculatng the covarant dervatve n the thrd bracket, ubttutng the exreon,, nto 0 and equlzng the rght de of the equaton 0 and, after longer arrangng one obtan U j v u W u W u m n n m {u U jv u A v mn W [u Γ [m] u β <β W,n v β Γ m Γ β n Γ m Γ β n β Γ [j β m] u v Γ m Γ nj β Γ m Γ j β n β v β <β Γ mnw ][mn] β Γ j m Γ nj β Γ mj Γ j β n }. W W,n W W j W Becaue U v j u an arbtrary tenor of the tye,, the lat equaton, n vew of,4, become W u v β m n W u u n m W W <[mn]> W A mn W [mn] Γ [mn] W.e. 0 vald for the tenor W of the tye u, v too, and Theorem roved. The followng theorem Th. - Th. 8 are roved n a mlar way..4. To the fourth cae from 4 related Theorem. Alyng two knd of covarant dervatve n the nvered order of that n the revou cae, we obtan the fourth Rcc tye dentty n O N for,

8 80 S. M. Mnčć bac dfferentaton: 4 5 V u m n V u n m mn β 4 V u <[mn]> V u [mn] Γ [mn] V u, jmn Γ mj,n Γ nj,m Γ mj Γ n Γ nj Γ m, 6 4 jmn Γ mj,n Γ nj,m Γ jm Γ n Γ jn Γ m..5. Further, we have Theorem. In O N vald the 5 th Rcc tye dentty for bac dfferentaton: 7 V u m n V u n m we have degnated 5 mn β 6 V u <mn> V u mn Γ mn, 8 5 jmn Γ jm,n Γ nj,m Γ jm Γ n Γ nj Γ m, 9 6 jmn Γ jm,n Γ nj,m Γ mj Γ n Γ jn Γ m. 40 u V u mn β <β β v Γ m Γ β n Γ m Γ β n Γ m Γ j β n Γ m Γ nj β Γ j m Γ j β n Γ mj Γ nj β β <β β j, whle m, n degnate the ymmetraton of the correondng exreon over m, n wthout dvon wth.

9 Rcc tye dentte for bac dfferentaton Theorem 4. In O N n force the 6 th Rcc tye dentty for bac dfferentaton: 4 4 V u m n V u n m 7 mn β V u <mn> V u mn, 8 7 jmn Γ jm,n Γ jn,m Γ jm Γ n Γ jn Γ m, 4 8 jmn Γ jm,n Γ jn,m Γ mj Γ n Γ jn Γ m. 44 u V u j j v mn β <β β v Γ Γ [m] Γ β n Γ n Γ β [m] [m] β Γ j β n Γ n Γ [j β m] Γ [j m] Γ j β n Γ j n Γ [j β m] j β <β..7. Theorem 5. In O N vald the 7 th Rcc tye dentty for bac dfferentaton: 45 V u m n V u n m we have degnated 9 mn β 0 V u <nm> V u j j v nm Γ mn Γ nm, 46 9 jmn Γ jm,n Γ nj,m Γ jm Γ n Γ nj Γ m, 47 0 jmn Γ jm,n Γ nj,m Γ jm Γ n Γ jn Γ m.

10 8 S. M. Mnčć.8. To the followng cae from 4 related Theorem 6. In O N vald the 8 th Rcc tye dentty for bac dfferentaton: 48 V u m n V u n m mn β V u j V u j v<nm> j j v mn Γ mn Γ nm, 49 jmn Γ mj,n Γ jn,m Γ mj Γ n Γ jn Γ m, 50 jmn Γ mj,n Γ jn,m Γ mj Γ n Γ nj Γ m. 5 u V u j j v mn.9. Alo we have β <β β v Γ m Γ β [n] Γ [n] Γ β m Γ m Γ [nj β ] Γ [n] Γ mj β β j Γ mj Γ [nj β ] Γ ] [nj Γ mj β β <β. Theorem 7. In an Otuk ace O N n force the 9 th Rcc tye dentty for bac dfferentaton: 5 V u m n V u n m mn β V u <mn> V u nm, 4 5 jmn Γ mj,n Γ nj,m Γ mj Γ n Γ nj Γ m,

11 Rcc tye dentte for bac dfferentaton jmn Γ mj,n Γ nj,m Γ jm Γ n Γ nj Γ m..0. For the lat cae from 4 we have Theorem 8. In O N vald the 0 th Rcc tye dentty for bac dfferentaton: 55 V u m n V u n m 5 mn Γ nm, β 5 56 θ A 5 jmn θ Γ jm,n θ Γ nj,m θ Γ jm θ Γ n θ Γ nj θ Γ m, θ,. The equaton 55 can be wrtten n another form. Namely, countng the dfference n the lat addend, we obtan another form of the 0 th Rcc dentty for bac dfferentaton n O N : 57 V u m n V u n m R mn β R 58 θ R jmn θ A 5 jmn Γ nm θ Γ [j], θ,. the curvature tenor of the rd knd n O N, determned by the connecton θ Γ, θ,. Remark. The quantte θ A t jmn θ, ; t,, 5 are not tenor and we call them curvature eudotenor of the ace O N of the t to 5 th knd reectvely. For examle, from the artcular cae of 4 V j m n V j n m 8 jmn V Γ [mj] V,n, we ee that 8 not a tenor, becaue V,n V / x n not a tenor.

12 84 S. M. Mnčć. The Rcc tye dentte for bac dfferentaton of the thrd and fourth knd One can defne n O N two new knd of bac covarant dervatve n lace of 7: V u V u m j j v,m Γ m Γ 59 mj β, 60 V u 4 m V u,m From here, t follow that Γ m β β Γ j β m. 6 δ j m Γ jm Γ mj, δ j 4 m Γ mj Γ jm. Analogouly to 4, we can obtan 0 new Rcc tye dentte n O N. For examle, V u V u R m n mn 6 n m β R Γ [mn] V u, V u V u β 4 m 4 n V u 4 n 4 m β m 4 n V u 4 Γ [mn] V u, V u <mn> R n 4 m R mn Γ [mn] V u 4, A mn V u j j v <[mn]> V u [mn] Γ [m],n β Γ [mj β ],n,

13 Rcc tye dentte for bac dfferentaton V u u mn β <β β v β <β Γ m Γ β n Γ m Γ β n Γ m Γ j β n Γ m Γ nj β Γ mj Γ j β n Γ j m Γ nj β β j. In thee dentte aear the ame quantte θ R, θ R, θ R ; θ A,, θ A 5, but n dfferent dtrbuton than n the cae.-.0. Only n the one cae aear a new curvature tenor R 4 : 67 V u m 4 n V u 4 n m R 4 mn β R j β nm, 68 R 4 jmn 5 jmn Γ mn Γ [j]. 4. Derved curvature tenor. Indeendent curvature tenor n O N A we have een, the quantte θ A θ, ; t,, 5 are not tenor. t We roved n [] for a non-ymmetrc connecton Γ Γ Γ that from the curvature eudotenor A one can obtan new, o called derved curvature t tenor. We can do th n an analogou way n O N too. For examle, addng the equaton 0 and 4, we get 69 m n n m m n R mn Γ [mn], β R n m

14 86 S. M. Mnčć 70 θ R jmn θ A θ A jmn θ A θ A 4 jmn, a tenor. Addng the equaton 4 and 5 we obtan 7 m n n m m n R mn R j β mn mn nm. β By vrtue of 44, 5 we ee that the quantty n m 7 V u j V u j v mn j j v mn u Γ [m] Γ β n Γ n Γ β [m] Γ n Γ β [m] β <β Γ [m] Γ β n β β Γ [m] Γ j β n Γ n Γ [j β m] Γ n Γ [mj β ] Γ [m] Γ nj β v Γ m] [j Γ j β n Γ j n Γ [j β m] Γ nj Γ [mj β ] β <β Γ [mj ] Γ nj β u β <β β v β <β j Γ [m] Γ β [n] Γ [n] Γ β [m] Γ [m] Γ [j β n] Γ [n] Γ [j β m] β Γ [j m] Γ [j β n] Γ [j n] Γ [j β m] j

15 Rcc tye dentte for bac dfferentaton a tenor, and n 7 the quantte 7 θ R jmn θ A 7 θ A jmn, θ R jmn θ A 8 θ A 4 jmn are alo tenor. We call the quantte θ R, θ R, θ R θ, derved curvature tenor of the ace O N. In addton to thoe reented here, one can obtan ome other uch tenor too ee []. By a rocedure analogou to that from [] t can be roved that from the curvature tenor θ R,..., θ R, θ R, θ R, θ R for a fxed θ or only fve of them 4 are ndeendent, whle the ret can be exreed a lnear conbnaton of thee fve tenor. If Γ Γ Γ, Γ a ymmetrc connecton, then all Rcc tye dentte reduce to the known Rcc dentty, all cted curvature tenor and eudotenor are reduced to the Remann-Chrtoffel curvature tenor of th connecton, whch can be ealy roved from the obtaned formula. Reference [] Mnčć, S., Curvature tenor of the ace of non-ymmetrc affne connexon, obtaned from the curvature eudotenor, Matem. venk, 8, 976, 4 45 [] Mnčć, S., Indeendent curvature tenor and eudotenor of ace wth nonymmetrc affne connexon, Coloq. math. Soceta Jáno Bolya,. Dff. geometry, Budaet Hungary, 979, [] Moór, A., Otukche Übertragung mt rekurrentem Maβtenor, Acta Sc. Math., 40, 978, 9 4 [4] Moór, A., Otukche Räume mt enem zwefach rekurrenten metrchen Grundtenor, Perodca Math. Hungarca, vol, 98, 9 5 [5] Nadj-F., Dj., On curvature of the Wel-Otuk ace, Publcatone Mathematcae, T , Fac., 59 7 [6] Otuk, T., On general connecton I, Math. J. Okayama Unv., 9, , [7] Prvanovć, M., On a ecal connecton of an Otuk ace, Tenor, N. S., vol 7, 98, 7 4 [8] Prvanovq M., Protrantvo Ocuk-Nordena, Izv. VUZ, Matematka 7, 984, 59 6 Receved by the edtor January, 00