RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES

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1 Mskolc Mathematcal Notes HU e-issn 787- Vol. 6 (05), No., pp DOI: 0.85/MMN RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES SVETISLAV M. MINČIĆ, MIĆA S. STANKOVIĆ, AND MILAN LJ. ZLATANOVIĆ Receved 9 October, 0 Abstract. Generalzed Fnsler space GF n s a dfferentable N -dmensonal manfold wth nonsymmetrc basc tensor g j.x; Px/ defned by condton (.). Usng the basc tensor, by (.) a non-symmetrc connecton P s defned, and also four knds of covarant dervatve n the Rund s sense and fve curvature tensors are obtaned (Secton ). In Secton two knds of Rcc coeffcents of rotaton are defned and ther propertes are exposed. Also, ntegrablty condtons of the equaton expressng covarant dervatves of the congruence vector by means of coeffcents of rotaton, are obtaned. In Secton a geodesc mappng of two spaces GF n and GF n s defned and some t s propertes are proved. In Secton some nvarants of such mappngs n relaton wth the coeffcents of rotaton are studed. 00 Mathematcs Subject Classfcaton: 5A5, 5B05, 5B0 Keywords: generalzed Fnsler spaces, non-symmetrc connectons, Rcc coeffcents of rotaton, geodesc mappngs, projectve varants. INTRODUCTION The generalzed Fnsler space.gf n / s a dfferentable manfold wth non-symmetrc basc tensor feld g j.x ;:::;x N ; Px ;:::; Px N / g j.x; Px/; where g j.x; Px/ g j.x; Px/;.g D det.g j / 0; Px D dx=dt/: (.) Based on (.), the symmetrc and ant-symmetrc part of g j are defned g j D.g j C g j /; Followng [8], n our notaton, hold g j D.g j g j /: a/ g j.x; Px/ Px/ Px j ; b/ D 0; Px k Research supported by Mnstry of Educaton Scence and Technologcal Development, Republc of Serba, Grant No. 70. c 05 Mskolc Unversty Press

2 06 SVETISLAV M. MINČIĆ, MIĆA S. STANKOVIĆ, AND MILAN LJ. ZLATANOVIĆ where F.x; Px/ s a metrc functon n GF n, havng the propertes known from the theory of usual Fnsler spaces.f n /. A lot of research papers (see for example [,6,, 9,,]) are dedcated to the theory of Fnsler spaces and ther generalzatons. The dscussed structure s a partcular case of the Esenhart-Lagrange approach whch s studed n [] (Chap. 8). The lowerng and the rasng of ndces are defned by the tensors g j and h j respectvely, where h j satsfy equaton g j h D ı k ;.g D det.g j / 0/: Generalzed Crstoffel symbols of the st and the nd knd are defned: : D.g j ;k g ; C g k;j / :kj ; D hp p: D hp.g jp;k g ;p C g pk;j / kj ; where, e.g., g j ;k j =@x k : Then we have p g p D s: h ps g p D s: ı s D : : Introducng a tensor C lke as at space F n, we have C.x; Px/ def D g j; Px k D g j ; Px k D F Px Px j Px k : We see that C s symmetrc n relaton to each par of ndces. Also, we have Wth help of coeffcents C def D h p C p D h p C jpk D h p C p : (.) P D C jp p sk Pxs P kj one obtans coeffcents of a non-symmetrc affne connectons n the Rund s sence (see [7, 8]): P D P : D P r g r D : h q.c jqp ks C C kqp js.c jp ks C C kp js C p qs / Pxs P kj ; (.) C p s / Pxs P :kj : In GF n we denote ant-symmetrc and symmetrc part for a connecton P respectvely: a/ T.x; Px/ D.P P kj / D. kj /; b/ P D.P C P kj /; where T s the torson tensor of the connecton P.

3 RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES 07 We defne four knds of covarant dervatve of a tensor n the space GF n. For example, for a tensor a j.x;/: a j jm.x;/ D a j;m C a j; Pp p ;m C m ap j a j jm.x;/ D a j;m C a j; Pp p ;m C P mp ap j a j jm.x;/ D a j;m C a j; Pp p ;m C m ap j a j jm.x;/ D a j;m C a j; Pp p ;m C P mp ap j j m a p ; mj a p ; (.5) mj a p ; j m a p ; (.6) where.x/ s an arbtrary tangent vector n the tangent space T n.x/, and a j; Pp =@ Pxp. In the work [0] we obtan 0 Rcc type denttes n the general case for a tensor a r :::r u t :::t v.x;/ and three curvature tensors of GF n : j mn D P j m;n j mn D P mj;n j mn D P j m;n P j n;m C j m P pn P nj;m C mj P np P nj;m C j m P np j n m C j m;ps s ;n j n;ps s ;m ; (.7) nj P mp C mj;ps s ;n nj;ps s ;m ; (.8) nj P pm C P nm p.p pj P jp / C j m;ps s ;n nj;ps s ;m : (.9) The magntudes t j mn ; t D ;; are tensors and we call them curvature tensors of the frst, the second and the thrd knd respectvely. In the work [9] we use the thrd and the fourth knd of covarant dervatve (.), and n that manner one gets 0 new Rcc type denttes. In these denttes appear the same quanttes t j mn ; t D ;;I ea t j mn ; t D ;:::;5, but n dfferent dstrbuton. Only n the last case appears a new curvature tensor (of the fourth knd) W where j mn D P j m;n a j jmjn C Pmn p.ppj a j jnj P nj;m C j m P np m D pmn a p j C p j nm a p ; nj m C j m;ps s ;n nj;ps s ;m Pjp /: (.0) Denotng by semcolon (;) the covarant dervatve wth respect to the symmetrc connecton P, then usng (.) we have j mn D j mn C T j min T j nim C T p j m T pn T p j n T pm ; (.)

4 08 SVETISLAV M. MINČIĆ, MIĆA S. STANKOVIĆ, AND MILAN LJ. ZLATANOVIĆ j mn D j mn C T mj In j mn D j mn C T j min j mn D j mn C T j min T nj Im C T p mj T np T nj Im C T p j m T np T nj Im C T p j m T np T p nj T mp ; (.) T p nj T pm C T p mn T jp ; (.) T p nj T pm T p mn T jp ; (.) where j mn s the curvature tensor formed by symmetrc connecton P j mn D P j m;n P j n;m C j m P pn j n P pm C P j m;ps s ;n P j n;ps s ;m : In the work [7] we fnd fve lnearly ndependent curvature tensors ;:::; 5, where ;:::; gven by equatons (.7-.0), (.-.), and 5 j mn D j mn C T p j m T pn C T p j n T pm ; (.5) j mn D.Pj m;n 5 P j n;m C j m P pn C mj P np nj P pm C j m;ps s ;n where j m denotes symmetrzaton by ndces j and m. Applyng two knds of covarant dfferentaton (.5), we get a j j m.x;/ D a j;m C a j; Pp p ;m C P pma p j mp j n P mp j n;ps s ;m /; (.6) j map mj D aj;m C a j; Pp p ;m C.Ppm T pm /ap j D a j Im C T pma p j mp T p j m mj a p :. j m T p j m /a p. RICCI COEFFICIENTS OF ROTATION IN A GENERALIZED FINSLER SPACE On Rcc coeffcents of rotaton n R n the reader can fnd e.g. n [0], 5. On that matter n GR n s wrtten n [8]. On Rcc coeffcents of rotaton n F n see []... Congruence of curves and orthogonal ennuple Defnton. A congruence of curves n a GF n s such a famly of curves that though each pont of GF n passes one curve of the famly. N mutually orthogonal congruences of curves consttute an orthogonal ennuple. Instead of congruences of curves we shall sometmes speak about congruences of the correspondng tangent vectors.

5 RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES 09 If.h/ ;.h D ;:::;N ) are unt tangent vectors of congruences of curves of an orthogonal ennuple, then n vrtue of the prevous defnton or g j.x; Px/ ị h/ j.k/ D e.k/ı hk ; e.k/ D ; (.) e.k/ ị h/.k/ D ı hk ; (.) where ı hk are Kronecker symbols. (of course, we do not suppose summaton w.r.t. (k) n (.), (.) and n smlar formulas later on.) The next theorem expresses the basc propertes of orthogonal ennuples. Theorem. For the unt tangent vectors.h/ ;.h D ;:::;N / of congruences of curves of an orthogonal ennuple the relatons NX N a/ e.k/.k/ j.k/ D ıj ; b/ X e.k/.k/.k/j D g j are vald. c/ kd kd (.) NX e.k/ ị k/ j.k/ D gj kd Proof. In the determnant det. ị k/ /, whose value s, we can regard e.k/.k/ as cofactor of the element ị : Developng the determnant ether by rows or by k/ columns.:a/ follows. Further, we have X X e.k/.k/.k/j D g jl e.k/.k/ ḷ k/ D g jl ı l ).:b/: k The equaton.:c/ can be obtaned n the same manner... Defnton and basc propertes of the coeffcent of rotaton k Usng the two knds of covarant dervatve (.5) of a vector n a GF n, we can defne two knds of coeffcents of rotaton, as two systems of nvarants (for D ;/. Defnton. The nvarants.hkm/.x; Px/ def D.h/ j j ị k/ j.m/ D ị h/ j j.k/ j ; D ;: (.).m/ are sad to be coeffcents of rotaton of the gven orthogonal ennuple. From (.5,.6) t s evdently that ị h/jj D ị h/jj ;.h/ j D.h/ j j ; j ị h/jj D ị h/jj ;.h/ j D.h/ j j ; j and t s easy to prove that there exst two knds of coeffcents of rotaton n the non-symmetrc case.

6 00 SVETISLAV M. MINČIĆ, MIĆA S. STANKOVIĆ, AND MILAN LJ. ZLATANOVIĆ Theorem. Both knds of coeffcents of rotaton are antsymmetrc n ther frst two ndces,.e..hkm/ D.khm/ ).hkm/ D 0: (.5) Proof. By covarant dfferentaton we get from (.) the relaton from where, transvectng by j.m/, that s e.k/. ị h/ j j.k/ C ị h/.k/ j j / D 0; e.k/. ị h/ j j.k/ j.m/ C ị h/.k/ j j j.m/ / D e.k/..:/.hkm/ C.khm/ / D 0 Theorem. If.hkm/ C.khm/ D 0 ).:5/:.hkm/ D.h/Ij ị k/ j.m/ (.6) are coeffcents of rotaton n the assocated Fnsler space F n.see []/, then.hkm/ D.hkm/ C. / T p j.h/p ị k/ j ; D ;; (.7).m/.hkh/ D.hkh/ ; Proof. In vrtue of (.) and we have.hkm/ D..hkm/ C.hkm//=;.hhk/ D.hhk/ ; j n D In C. / T p n p; D ;;.hkk/ D.hkk/ ; D ;: (.8).hkm/ D..h/Ij C. / T p j.h/p/ ị k/ j.m/ ).:7/ D ;:.:6/ In vrtue of (.7) for two ndces there follows (.8) because for example (for h D m) s T p j.h/p ị k/ j.m/ D T p:j p.h/ ị k/ j.h/ D T j:p j.h/ ị k/ p.h/ D T p:j p.h/ ị k/ j.h/ D 0; where T p:j D T s j g sp. Here we appled the fact that T p:j s antsymmetrc n all pars of ndces.

7 RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES 0.. Expresson of the dervatve of the vectors of a congruence by the coeffcents of rotaton Theorem. In GF n the relaton NX a/.h/ j D e.k/ e.m/ j.hkm/.k/.m/j ; are vald. b/ ị h/ j j D k;md N X k;md e.k/ e.m/.hkm/ ị k/.m/j (.9) Proof. a/ Multplyng the relaton (.) by e.k/ e.m/.k/p.m/q and summng wth respect to m;k get X e.k/ e.m/.hkm/.k/p.m/q D X.h/ j j ị k/ j.m/ e.k/e.m/.k/p.m/q k;m k;m D.h/ j j f X k e.k/.k/p ị k/ g fx m D.h/ j j ıp ıj q D.h/p j q:.:a/ e.m/.m/q j.m/ g Theorem 5. The covarant dervatves of vectors.h/ and ị n the drecton of h/ the vector j can be expressed by the coeffcents of rotaton as lnear combnaton.p/ of the vectors of the ennuple as follows: a/.h/ j j j.p/ D X k e.k/.hkp/.k/ b/ ị h/ j j j.p/ D X k Proof. Transvectng the equaton (.9) by j, we obtan.p/ a/.h/ j j j.p/ D X e.k/ e.m/.hkp/.k/.m/j j.p/ k;m e.k/.hkp/ ị k/ (.0) D X k;m e.k/.hkp/.k/ ı mp D X k e.k/.hkp/.k/ ).:0/:.. Integrablty condtons of the equaton (.9) The relaton (.9) s a partal dfferental equaton wth respect to the unknown functons.h/. Now we are gong to examne ts ntegrablty condton.

8 0 SVETISLAV M. MINČIĆ, MIĆA S. STANKOVIĆ, AND MILAN LJ. ZLATANOVIĆ In [0] we have obtaned 0 Rcc-type denttes n GF n. In three of these denttes appear the curvature tensors,,, and n the others appear the quanttes ea ;:::; ea 5 whch have the form and the role of the curvature tensors, but they are not tensors. In [5] we obtaned combned Rcc-type denttes, n whch appear derved curvature tensors ;:::; 8 : In [7] s proved that only fve are ndependent among the mentoned curvature tensors, for example ; ; ; ; whle the others are lnear combnatons of these fve tensors and. We shall use further 5 those of the Rcc-type denttes n whch appear the above tensors (the tensor s lnear combnaton of ; ; whle the tensor does not appear n denttes whch we need.) Theorem 6. In GF n the frst two ntegrablty condtons ( D ;/ of equatons.:9/ are s. jr.h/s C. / T s jr.h/ j..hpt/;j C.hpt/;Ps s j /j.q/ C C.hpk/ Œ.kqt/ s/ ị p/ j.q/ ṛ t/ D..hpq/;j C.hpq/;Ps j s /j.t/ NX kd.ktq/ g; D ; where ; are gven by.:7/. e.k/ f.hkq/.kpt/ Proof. Applyng the Rcc-type denttes from [0], we have.h/ j s jr.h/ j D jr.h/s C. / rj T s jr.h/ j.hkt/.kpq/ s; D ;; (.) By repeated dfferentaton of (.9) one can form the dfference on the left sde of ths equaton, and then (.) easly follows. Theorem 7. The thrd ntegrablty condton of the equaton.:9/ n GF n s s jr.h/s ị p/ j.q/ ṛ t/ D..hpq/;j C.hpq/Ps s j /j.t/ N..hpt/;j C.hpt/Ps j s /j.q/ C X e.k/ Œ.hkq/.kpt/ C.hpk/.kqt/ where s gven by.:9/. kd.hpk/.ktq/ :.hkt/.kpq/ (.)

9 RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES 0 Proof. Applyng the correspondng dentty from [0] we get.h/ j j j r.h/ j r j j D s jr.h/s : Further, we use (.9) to form the dfference on the left sde. Theorem 8. The fourth ntegrablty condton of the equaton.:9/ s sjr ṣ h/.p/ j.q/ ṛ t/ D..hpq/;j C.hpq/Ps s j /j.t/ N..hpt/;j C.hpt/Ps j s /j.q/ C X e.k/ Œ.hkq/.kpt/ C.hpk/.kqt/ kd.hkt/.kpq/.hpk/.ktq/ : (.) where s gven by.:0/. Proof. Usng by equatons (.0) we see that ị h/jj jr ị h/jr j j D sjr ṣ h/ ; and the use of (.9) yelds the ntegrablty condton (.). Theorem 9. The ffth ntegrablty condton of the equaton.:9/ s sjr ṣ h/.p/ j.q/ ṛ t/ D..hpŒq/;j C.hpŒq/Ps s j /j.t/..hpœq/;j C.hpŒq/Ps s j /j.t/ C X k e.k/ Œ.hkq/.kpt/ C.hpk/.kqt/.hkt/.kpq/.hkt/.kpq/.hkt/.kpq/ C.hkq/.kpt/ C.hpk/.kqt/.hkt/.kpq/ ; where Œqt we denoted symmetrzaton wthout dvson by ndces q and t and sjr 5 sjr gven by.:6/. Proof. We use that Rcc-type dentty n whch the curvature appears. From correspondng denttes n [5] we have and then use (.9). ị h/jjr ị h/jr jj C ị h/jjr ị h/jr j j D sjr ṣ h/ ;

10 0 SVETISLAV M. MINČIĆ, MIĆA S. STANKOVIĆ, AND MILAN LJ. ZLATANOVIĆ. GEODESIC MAPPINGS Defnton. Geodesc n GF n s gven by d x ds C P dx dx k.x; ds /dxj ds ds D 0: (.) Consder two N -dmensonal spaces of non-symmetrcal affne connecton: GF n and GF n. So, f connecton coeffcents of these spaces are respectvely P and P, we suppose that n general the symmetry wth respect to ndces j;k s not n force. One says that recprocal one valued mappng f W GF n! GF n s geodesc, f geodescs of the space GF n pass to geodescs of the space GF n. We can consder these spaces n the common by ths mappng system of local coordnates,.e. for f W M 7! M we have M.x ;:::;x n ; Px ;:::; Px n / M.x; Px/ and M.x ;:::;x n ;x ;:::; x n ; Px ;:::; Px n / M.x; Px/, where M GF n, M GF n. In the correspondng ponts M.x; Px/ and M.x; Px/ we can put P D P C D ;.;j;k D ;:::;N /; (.) where D.x/ s the deformaton tensor of the connecton P of GF n accordng to the mappng f W GF n! GF n. Defnton. The curve s geodesc of GF n f and only f s: l W x D x.t/ where.t/ s an nvarant and t an arbtrary parameter. Rx C P pq Pxp Px q D.t/ Px.t/; (.) If f W l! l, then n the common by the mappng f coordnates x x t s Px D dx =dt D Px, and l s geodesc n GF n too, from where we get Subtractng (.) and (.), we obtan and, because of (.):.P pq Rx C P pq Pxp Px q D.t/ Px.t/ (.) P pq / Pxp Px q D..t/ the symmetrc and antsym- where.t/ D..t/.t//=. Denotng by D, D metrc part of D respectvely, we get.t// Px.t/; D pq Pxp Px q D.t/ Px.t/; (.5) D D D C D ;

11 and (.5) reduces to RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES 05 D pq Pxp Px q D.t/ Px.t/: (.6) As n the case of a symmetrc connecton (see for ex. []) one concludes that and from (.6):.t/ D p.x.t/;:::;x n.t/; Px.t/;:::; Px n.t// Px p.t/; D pq Pxp Px q D p Px p Px D p Px p Px q ı q C q Px p Px q ı p D. pı q C qı p / Pxp Px q ; wherefrom Denotng also by substtuton n (.) we obtan and the deformaton tensor s D D ı j k C ı k j : (.7) D D D kj ; P D P C ı j k C ı k j C ; (.8) D D ı j k C ı k j C : (.9) So, the condton (.8) s necessary that the mappng f be geodesc. It s easly to prove that ths condton s suffcent too, and we have Theorem 0. A necessary and suffcent condton that the mappng f W GF n! GF n be geodesc s the deformaton tensor D from.:/ at the mappng f to has the form.:9/, where j.x; Px/ s a covarant vector, and.x; Px/ an antsymmetrc tensor. For k D, we obtan from (.7): wherefrom D j D ı j C ı j D j C N j ; D N C Dp p ; (.0) whch, by substtuton n (.8), gves P D P C N C.ı j Dp kp.x; Px/ C ı k Dp jp.x; Px// C D.x; Px/; (.) where D.x/ s the deformaton tensor. On the base of above explaned, we get

12 06 SVETISLAV M. MINČIĆ, MIĆA S. STANKOVIĆ, AND MILAN LJ. ZLATANOVIĆ Theorem. Let a space GF n be gven,.e. on a dfferentable manfold M n let be defned non-symmetrc connecton coeffcents P.x; Px/. If on M n s gven a tensor D.x; Px/ too and we determne P.x; Px/ accordng to.:/, then on M n wll be defned a space GF n, wth connecton coeffcents P, and then GF n and GF n have common geodescs. We obtan the same result.on the base of.:8/ by choce of a vector.x; Px/ and antsymmetrc tensor.x; Px/ D D.x; Px/. The queston forces tself: Is t possble a geodesc mappng of a space GF n wth a non-symmetrc affne connecton on to a space F n wth a symmetrc affne connecton? It s easy to see that the next theorem s vald: Theorem. A necessary and suffcent condton that a mappng f W GF n! F n of a non-symmetrc affne connecton space GF n onto a symmetrc affne connecton space F n be geodesc, s D.x; Px/ D P.x; Px/; where D.x; Px/, P.x; Px/, are antsymmetrc parts of the deformaton tensor and connecton coeffcents of the GF n respectvely. Remark. It s easy to check that a set of geodesc mappngs of a space GF n makes a group.. SOME PROJECTIVE INVARIANTS OF GEODESIC MAPPINGS Puttng D nto (.) n accordance wth (.) we get P Denotng P we see that N C.ı j kp C ı k jp / D P T D P P N C.ı j kp C ı k jp /; N C.ı j kp C ı k jp / D T kj ; (.) T D T : (.) The magntudes T we call generalzed Thomas s projectve parameters at the mappng f W GF n! GF n. Accordngly, these magntudes are nvarant at a geodesc mappng. Startng from (.,.), one obtans (.), and we conclude that the next theorem s vald: Theorem. A necessary and suffcent condton that a mappng f W GF n! GF n be geodesc s that generalzed Thomas s projectve parameters.:/ are nvarant, that s.:/ to be vald.

13 RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES 07 Theorem. If ị h/.x/ and.h/.x/ are the contravarant, respectvely covarant components of an orthogonal ennuple, then the followng geometrc enttes are nvarant under the geodesc mappng: M k.x; Px/ D ị h/ j.x/ n C p.h/.x/x h e.h/.h/q ı q.p.h/ j k/; D ;; (.) and M k.x; Px/ D X e.h/.h/p p D q ; D ;; (.).h/ j k kq h where.p :::k/ denotes symmetrzaton by respect of ndces p and k. Proof. If we denote by ị ; D ; the covarant dervatve wrt the connecton h/ j k P n sense of (), we have Usng (.8,.) we have ị h/ ị h/ ị h/j D k ị h/;k C p.h/ P pk: kp (.5) ị h/ D p.h/.ı p k C ı k p C pk / ị a/ D p.h/.ı k p C ı p k C kp /: (.6) Multplyng (.6) by.h/ and summng wth respect to ndces h and usng orthogonalty condton, we have X h X h e.h/.h/p. p.h/ e.h/.h/p. p.h/ p.h/ / D.N C / k p.h/ / D.N C / k: Elmnatng the k from (.6) and (.7) we get 0 ị k h/ D n C X e.h/.h/.h/q ı q.p.h/j h ị h/j Wth help (.0) we get (.). k/ X e.h/.h/q ı q.p.h/j h k/ (.7) Theorem 5. When GF n and GF n are n geodesc correspondence, we have the followng projectve nvarant enttes: S hkm.x; Px/ D.hkm/ n C.ı hk ḳ m/ P k C ı km j.h/ P j /; D ;; A:

14 08 SVETISLAV M. MINČIĆ, MIĆA S. STANKOVIĆ, AND MILAN LJ. ZLATANOVIĆ where.hkm/ D ;; are Rcc coeffcents of rotaton. Proof. Usng (.5) we get ị h/ ị h/j k D N C j.h/.ı j Dp kp.x; Px/ C ı k Dp jp.x; Px//: (.8) Multplyng (.8) by.k/ ḳ and usng orthogonalty condton, we have m/ hkm n C.ı hk ḳ m/ P k C ı km q.h/ qp / D hkm n C.ı hk ḳ m/ kp C ı km j.h/ jp /; where the projectvely transformed Rcc coeffcents of rotaton are gven by hkm D.h/p j q p.k/ q.m/ D p.h/ j q.k/p q.m/ ; D ;: ACKNOWLEDGEMENT The authors are grateful to the Professor Vladmr Balan for valuable comments and suggeston references, whch were most helpful n mprovng the manuscrpt. REFERENCES [] V. Balan and N. I. Rodca, Berwald-moor metrcs and structural stablty of conformally deformed geodesc SODE, Appl. Sc., vol., pp. 9, 009. [] L. P. Esenhart, Generalzed Remannan spaces, Proc. Nat. Acad. Sc. USA, vol. 7, pp. 5, 95, do: 0.07/pnas [] H. Goenner, On the Hstory of Unfed Feld Theores. Part, ed. Lvng Revews n Relatvty: Lvng Rev. Relatvty, 0, vol. 7. [] J. Mkeš, S. Báscó, and V. Berezovsk, Geodesc mappngs of weakly Berwald spaces and Berwald spaces onto Remannan spaces, Int. J. Pure Appl. Math., vol. 5, no., pp. 8, 008. [5] S. M. Mnčć, Curvature tensors of the space of non-symmetrc affne connexon, obtaned from the curvature pseudotensors, Mat. vesnk, vol..8/, no., pp. 5, 976. [6] S. M. Mnčć, New commutaton formulas n the non-symmetrc affne connecton space, Publ. Inst. Math. (Beograd) (N. S), vol..6/, pp , 977. [7] S. M. Mnčć, Independent curvature tensors and pseudotensors of spaces wth non-symmetrc affne connecton, Coll. Math. Soc. János Bolya, vol., pp. 5 60, 979. [8] S. M. Mnčć, Rcc coeffcents of rotaton n a generalzed Remannan space, Publ. Math. Debrecen, vol., no. -, pp. 7 80, 99. [9] S. M. Mnčć and M. L. Zlatanovć, New commutaton formulas for ı-dfferentaton n generalzed Fnsler space, Dffer. Geom. Dyn. Syst., vol., pp. 5 57, 00. [0] S. M. Mnčć and M. L. Zlatanovć, Commutaton formulas for ı-dfferentaton n a generalzed Fnsler space, Kragujevac J. Math., vol. 5, no., pp , 0.

15 RICCI COEFFICIENTS OF ROTATION OF GENERALIZED FINSLER SPACES 09 [] S. M. Mnčć and M. L. Zlatanovć, Geometrc nterpretaton of the torson tensor, curvature tensors and pseudotensors n generalzed Fnsler space (n Russan), Izv. Vyssh. Uchebn. Zaved. Mat., vol., pp. 0, 0. [] R. Mron and M. Anastase, Vector bundles and Lagange spaces wth applcatons to relatvty. Geometry Balkan Press, 997. [] H. D. Pande, Projectve nvarants of an orhogonal ennuple n a Fnsler space, Annales de l nsttut Fourer, vol. 8, no., pp. 7, 968, do: 0.580/af.0. [] H. D. Pande and K. K. Gupta, Banch s denttes n a Fnsler space wth non-symmetrc connecton, Bull. Acad. Serbe Sc. Arts Cl. Sc. Math. Natur. (N.S.), vol., no. 0, pp. 6, 979. [5] S. Rastog, On projectve nvarants based on non-lnnear connnecton n a Fnsler space, Publ. Math. Debrecen, vol. 8, no., pp. 8, 98. [6] S. Rastog, On projectve nvarants based on non-lnnear connnecton n a Fnsler space, Internatonal Centre For Theoretcal Physcs, Treste, Italy, pp. 8, 986. [7] H. Rund, The Dfferental Geometry of Fnsler Space (n Russan). Moskow, 98. [8] A. C. Shamhoke, Some propertes of a curvature tensors n a generalsed Fnsler space, Tensor (N.S.), vol., pp , 96. [9] A. C. Shamhoke, A note on a curvature tensor n a generalzed Fnsler space, Tensor, N.S., vol. 5, pp. 0, 96. [0] C. E. Weatherburn, Remannan geometry and tensor calculus. Cambrdge U. P., 950. [] M. L. Zlatanovć and M. Cvetkovć, New Cartan s tensors and pseudotensors n a generalzed Fnsler space, Flomat, vol. 8, no., pp. 07 7, 0. [] M. L. Zlatanovć and S. M. Mnčć, Identtes for curvature tensors n generalzed Fnsler space, Flomat, vol., no., pp., 009, do: 0.98/FIL0900Z. Authors addresses Svetslav M. Mnčć Unversty of Nš, Faculty of Scence and Mathematcs, Všegradska, 8000 Nš, Serba E-mal address: svetslavmncc@yahoo.com Mća S. Stankovć Unversty of Nš, Faculty of Scence and Mathematcs, Všegradska, 8000 Nš, Serba E-mal address: stmca@ptt.rs Mlan Lj. Zlatanovć Unversty of Nš, Faculty of Scence and Mathematcs, Všegradska, 8000 Nš, Serba E-mal address: zlatmlan@yahoo.com