Geodesic mappings of equiaffine and anti-equiaffine general affine connection spaces preserving torsion

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1 Flomat 6: (0) 9 DOI 0.98/FIL09S Publshed by Faculty of Scences and Mathematcs Unversty of Nš Serba Avalable at: Geodesc mappngs of equaffne and ant-equaffne general affne connecton spaces preservng torson Mća S. Stankovć a Marja S. Ćrć a Mlan Lj. Zlatanovć a a Unversty of Nš Faculty of Scence and Mathematcs Všegradska 8000 Nš Serba Abstract. In ths paper we consder equtorson geodesc mappngs of equaffne spaces of the -knd {0... }. In the case when {0 } vector ψ whch determnes that mappng s gradent whch doesn t hold n general case. We found the condton when the vector ψ s gradent n the case of {... }. Some nvarant geometrc objects of such mappngs are found. Ant-equaffne spaces of -knd {0... } are ntroduced and dscussed. Equtorson geodesc mappngs of such spaces are descrbed and some nvarants are found.. Motvaton The study of the theory of geodesc mappngs of emannan spaces affne connecton spaces and ther generalzatons has been an actve feld over the past several decades. Many new and nterestng results appeared n the papers of N. S. Snyukov [6] J. Mkeš [7]-[0] G. S. Hall and D. P. Lone []-[6] M. Prvanovć [ ] S. Mnčć [ ] Z. adulovć [] N. Pušć [ ] etc. The nvestgaton of geodesc mappngs theory for specal spaces s an mportant and actve research topc. The begnnng of the study of general (non-symmetrc) affne connecton spaces s especally related to the works of A. Ensten [ ] on Unfed Feld Theory (UFT). Ensten was not satsfed wth hs General Theory of elatvty (GT 96) and from 9 to the end of hs lfe (9) he worked on dfferent varants of UFT. Ths theory had the am to unte the gravtaton theory to whch s related GT and the theory of electromagnetsm. At the second step L. P. Esenhart [] nvestgated the propertes of generalzed emannan spaces enabled wth non-symmetrc metrcs. He put the problem to defne the class of lnear connectons whch are compatble wth the symmetrc part of a non-symmetrc metrc. Further developments extended the problem of generalzaton to noncommutatve quantum gravty and applcatons n modern cosmology. Equaffne spaces are determned by the symmetry of the cc tensor. These spaces wth symmetrc affne connecton n whch the volume of an N-dmensonal parallelepped s nvarant under parallel transport play an mportant role n the theory of geodesc mappngs. Geodesc mappngs of equaffne spaces wth symmetrc affne connecton were studed n [0]-[]. For nstance n [] t s proved that a 00 Mathematcs Subject Classfcaton. Prmary A; Secondary B0 Keywords. Equtorson geodesc mappng general affne connecton space generalzed emannan space equaffne space ant-equaffne space eceved: 07 July 0; Accepted: September 0 Communcated by Dragan S. Djordjevć esearch supported by Mnstry Educaton and Scence epublc of Serba Grant No. 70 Emal addresses: stmca@ptt.rs (Mća S. Stankovć) marjamath@yahoo.com (Marja S. Ćrć) zlatmlan@yahoo.com (Mlan Lj. Zlatanovć)

2 M.S. Stankovć et al. / Flomat 6: (0) 9 0 manfold wth symmetrc affne connecton s locally projectvely equvalent to an equaffne manfold. In [] more general theorem s gven: All manfold wth affne connecton are projectvely equaffne. In [0] fundamental equatons of geodesc mappngs of equaffne spaces nto (pseudo-) emannan spaces were found. Four knds of covarant dervatves [] fve ndependent curvature tensors [7] and fve cc tensors exst n the spaces wth non-symmetrc affne connecton. Also there exsts one more cc tensor whch we take from the adjont symmetrc space. If one of them s symmetrc t doesn t mean that all other cc tensors are symmetrc too. In ths way we can defne sx types of equaffne spaces wth non-symmetrc affne connecton. We consder geodesc mappngs of such spaces. Also f we suppose ant-symmetry of any cc tensor we get ant-equaffne space. Geodesc mappng of ant-equaffne spaces are especally nterestng. Equtorson mappngs are specal geodesc mappngs of non-symmetrc affne connecton spaces. These mappngs are ntroduced by M. Stankovć at [8] and they represent geodesc mappngs whch preserve torson of the spaces. The paper s organzed n the followng way. In Secton some used notatons and prelmnares are ntroduced. In Secton equaffne general affne connecton spaces of the -knd {0... } are ntroduced and equtorson geodesc mappngs of such spaces are consdered. It s proved that n the case when {0 } vector ψ whch determnes that mappng s gradent. For the rest of the cases vector ψ s not a gradent. Also we found the condton when the vector ψ s gradent n the case of {.. }. Some nvarant geometrc objects of such mappng are found. In Secton ant-equaffne general affne connecton spaces of -knd {0... } are ntroduced and dscussed. Equtorson geodesc mappngs of such spaces s descrbed and some nvarants were found. Specally t s proved that the ffth curvature tensor s nvarant of equtorson geodesc mappngs of ant-equaffne generalzed emannan spaces of -knd {... }. In Secton we gve a concludng remark of the paper.. Notatons and prelmnares A generalzed emannan space G N n the sense of Esenhart s defnton [] s a dfferentable N- dmensonal manfold equpped wth a non-symmetrc basc tensor j. The general (non-symmetrc) affne connecton space GA N s a dfferentable N-dmensonal manfold where ntroduced magntudes are L nstead of basc tensor j whch transform themselves by the low: L j k (x ) = L (x)x xj j x k k + x x j k (x = x x x = x x = x x j k x j x k (x ) and (x ) are two knds of local coordnates) where s L L kj (.) n the general case. The magntudes L are coeffcents of the non-symmetrc affne connecton. The space G N s specal case of the space GA N. Connecton coeffcents of the space G N are generalzed Crstoffel s symbols of the second knd Γ. Generally Γ Γ kj. Based on (.) one can defne the symmetrc part of L : L = (L + L kj ) = L () and ant-symmetrc part L = (L L kj ) = L []. The magntude L s torson tensor. Obvously L = L + L. (.)

3 M.S. Stankovć et al. / Flomat 6: (0) 9 The magntudes L can be consdered as the coeffcents of the symmetrc affne connecton space A N. Usng the non-symmetrc connecton n the space GA N one can defne four knds of covarant dervatves [ 6]. For example for a tensor a we have j a j a j m = a jm + L pma p j Lp jm a p a j m = a jm + L mpa p j Lp mj a p m = a jm + L pma p j Lp mj a p a j m = a jm + L mpa p j Lp jm a p. In the case of the space GA N we have fve ndependent curvature tensors [7]: jmn = L jmn L jnm + Lp jm L pn L p jn L pm jmn = L mjn L njm + Lp mj L np L p nj L mp jmn = L jmn L njm + Lp jm L np L p nj L pm + L p nm(l pj L jp ) jmn = L jmn L njm + Lp jm L np L p nj L pm + L p mn(l pj L jp ) (.) (.) jmn = (L jmn + L mjn L jnm L njm + Lp jm L pn + L p mj L nm L p jn L pm L p nj L pm). These curvature tensors produce cc tensors of -knd.e. α jmα = jm {... }. emanan curvature tensor formed by symmetrc part of connecton L exsts n the GA N: jmn = L jmn L jnm + Lp jm L pn L p jn L pm. (.) The cc tensor s gven by α jα = j. Corollary.. In the case when L = 0 all tensors gven by (.) reduce to one curvature tensor (.). Let GA N and GA N be two general affne connecton spaces. A dffeomorphsm f : GA N GA N s called geodesc mappng of GA N nto GA N f f maps any geodesc curve n GA N nto a geodesc curve n GA N [8]. In the correspondng ponts M(x) and M(x) we can put L (x) = L (x) + P (x) ( j k =... N) (.6) where P (x) s the deformaton tensor of the connecton L of GA N accordng to a mappng f. A necessary and suffcent condton that the mappng f can be geodesc [8] s that the deformaton tensor P has the form P = δ j ψ k + δ k ψ j + ξ (.7) ψ = + N Pp p = + N (Lp p Lp ) (.8) p ξ = P = L L. Defnton.. ([8]) A geodesc mappng f : GA N GA N s equtorson f the torson tensors of the spaces GA N and GA N are equal n the correspondng ponts. Accordng to (.9) t means that (.9) L L = ξ = 0. (.0)

4 . Equtorson geodesc mappng of equaffne spaces M.S. Stankovć et al. / Flomat 6: (0) 9 Defnton.. The general affne connecton space GA N s equaffne of the -knd {... } f the cc tensor of the -knd s symmetrc.e. jm = mj. The space GA N s zero-equaffne f the cc tensor s symmetrc.e. j = j. Let us consder an equtorson geodesc mappng of two equaffne spaces of the -knd {0.. }. By vrtue of the geodesc mappng f : GA N GA N we obtan tensors ( =... ) where for example jmn = L jmn L jnm + Lp jm L pn L p jn L pm. We demonstrate here how one can represent the tensor gven by equaton (.) usng (.): jmn = L jmn + L jmn L jnm L jnm + Lp jm L pn + L p L jm pn + L p jm L pn L p jn L pm L p L jn pm L p jn L pm L p L jn pm = jmn + L jm;n L jn;m + Lp jm L pn L p jn L pm + L p L jm pn (.) where (;) denotes covarant dervatve wth respect to the symmetrc connecton L and s emannan jmn curvature tensor gven by (.). Followng ths procedure we obtan: jmn = jmn L jm jmn = jmn + L jm jmn = jmn + L jm jmn = jmn + Lp jm ;n + L jn;m + Lp L jm pn ;n + L jn;m Lp L jm pn ;n + L jn;m Lp L jm pn L pn + L p jn L pm. L p jn L pm + L p L jn pm L p mn + L p L jn pm + L p mn L pj L pj Contractng by ndces and n n (.) and (.) we get fve cc tensors: jm = jm + L p jm jm = jm L p jm jm = jm + L p jm jm = jm + L p jm ;p Lp jp ;p + Lp jp ;p + Lp jp ;p + Lp jp jm = jm + L p L q jm pq + L p jq ;m + Lp L q jm pq ;m + Lp L q jm pq ;m Lp L q jm pq ;m Lp L q jm pq L q pm. L p jq L p jq L q pm L q pm + L p L q jq pm + L p L q jq pm L p mq + L p mq By alternatng wthout dvson wth respect to the ndces j and m n (.) we get [jm] = [jm] + L p jm [jm] = [jm] L p jm [jm] = [jm] + L p jm [jm] = [jm] + L p jm [jm] = [jm] + L p jm ;p Lp jp ;p + Lp jp ;p + Lp jp ;p + Lp jp ;m + Lp mp ;m Lp mp ;m Lp mp ;m Lp mp ;j + Lp jm ;j + Lp jm ;j Lp jm ;j Lp jm L q pj L q pj (.) (.) (.)

5 where [ ] denotes ant-symmetrzaton wthout dvson. M.S. Stankovć et al. / Flomat 6: (0) 9 emark.. In the space GA N s vald [jm] = [jm] where jm and jm are c tensors of the thrd and the fourth knd respectvely. In the case of geodesc mappng of two generalzed emannan spaces vector ψ gven by formula (.8) whch determnes the geodesc mappng s gradent [8]. Also ths fact s vald for geodesc mappng of two spaces of symmetrc affne connecton.e. the next theorem s vald: Theorem.. ([]) If the mappng f : A N A N s geodesc mappng of two equaffne spaces then the vector ψ s gradent. In the general case of geodesc mappngs of two general affne connecton spaces t s not vald. In that case the next theorems are vald: Theorem.. If the mappng f : GA N GA N s equtorson geodesc mappng of two equaffne spaces of the -knd {0 } then the vector ψ s gradent. Proof. Usng (.6) (.7) (.0) and (.) we get the relaton between the ffth curvature tensors of the spaces GA N and GA N under equtorson geodesc mappng: jmn = jmn + δ j (ψ mn ψnm + ψmn ψ where we denoted nm) + δ m(ψ jn + ψ jn) δ n(ψ jm + ψjm) (.) ψmn = ψ m n ψ m ψ n α { }. (.6) α α Contractng by ndces and n n (.) we get jm = jm + (ψ mj ψ jm + ψ mj ψjm) + (ψ jm + ψjm) N(ψ Alternatng by ndces j and m wthout dvson n (.7) we get [jm] = [jm] N + (ψ [jm] + ψ Accordng to (.) and (.6) we have ψ [jm] + ψ[jm] = (ψ jm ψ mj ). eplacng (.9) n (.8) one obtans jm + ψjm). (.7) [jm]). (.8) [jm] = [jm] (N + )(ψ jm ψ mj ). (.0) If we suppose that the tensors jm and jm are symmetrc t s vald ψ jm = ψ mj.e. ψ s gradent. Otherwse replacng (.) nto (.0) we get [jm] + L p jm = [jm] + L p L q jm pq ( + N)(ψ jm ψ mj ). As the spaces GA N and GA N have the same torson tensors under the equtorson mappng we have [jm] = [jm] ( + N)(ψ jm ψ mj ). (.) If the tensors jm and jm are symmetrc then ψ s gradent. (.9)

6 M.S. Stankovć et al. / Flomat 6: (0) 9 From the equaltes (.0) (.) we have the followng corollares: Corollary.. Let f : GA N GA N be equtorson geodesc mappng. The tensor [jm] s nvarant of ths mappng f and only f [jm] s nvarant of ths mappng. Corollary.. If f : GA N GA N s equtorson geodesc mappng of two equaffne spaces of the zero knd then the tensor [jm] s an nvarant of ths mappng. And nversely f f : GA N GA N s equtorson geodesc mappng of two equaffne spaces of ffth knd then the tensor [jm] s an nvarant of ths mappng. Under equtorson geodesc mappngs of two equaffne spaces of -knd {... } vector ψ sn t gradent n general case but t holds Theorem.. Let f : GA N GA N be equtorson geodesc mappng of two equaffne spaces of the -knd {... }. Vector ψ s gradent f and only f L p ψ mj p = N (Lα mαψ j L α jαψ m ). (.) Proof. Let us suppose that the spaces GA N and GA N are equaffne of the frst knd. Let us start from the frst curvature tensor. Usng (.6) (.7) (.0) and (.) we get the relaton between the frst curvature tensors of the spaces GA N and GA N under equtorson geodesc mappng jmn = jmn + δ j (ψ mn ψ nm) + δ mψ Contractng by ndces and n n (.) we get jm = jm + (ψ where we denoted mj ψ ψmn = ψ m n ψ m ψ n. jm) + ψ jn δ nψjm + L mnψ j + L α mnψ α δ j. (.) jm Nψjm + L α mα ψ j + L α mj Alternatng by ndces j and m wthout dvson n (.) we get [jm] = [jm] ( + N)ψ[jm] + L α mα As [jm] = 0 [jm] = 0 we get ψ j L α jα ψ m + L α mjψ α. ψ α (.) (.) ( + N)ψ[jm] = L α mαψ j L α jαψ m + L α mjψ α. (.6) Accordng to (.) and (.) we have ψ [jm] = ψ jm ψ mj + L p mj eplacng (.7) nto (.6) one obtans ψ jm ψ mj = +N ψ p. (.7) ( ( N)L p mj ψ p + L α mα ψ j L α jα ψ m ). From here we can see that ψ s gradent f and only f the equaton (.) holds. On the same way t s easy to prove that for the other curvature tensors equaton (.) s necessary and suffcent condton to ψ be gradent. The Corollares. and. don t hold for the other cc tensors n general case but they do n the case of specal equtorson geodesc mappngs. Namely

7 M.S. Stankovć et al. / Flomat 6: (0) 9 Theorem.. Let f : GA N GA N be equtorson geodesc mappng of two general affne connecton spaces whch satsfes L p ψ mj p = N (Lα mαψ j L α jαψ m ). (.8) Then the next condtons are equvalent: a) [jm] = [jm] b) [jm] = [jm] c) [jm] = [jm] d) [jm] = [jm] e) [jm] = [jm] f ) [jm] = [jm]. Proof. Let us prove the equvalence (a) (b).e. [jm] = [jm] [jm] = [jm]. Let (;) denote covarant dervatve wth respect to the symmetrc connecton L and (;) covarant dervatve wth respect to the L. Accordng to (.) and (.6) one obtans L jm ;k = L jm k + L pkl p jm and from (.) we get L p L pm L p mkl jp = L jmk + (L pk + P pk )Lp (L p jm + Pp )L pm (L p mk + Pp mk )L jp = L jm;k + P pk Lp P p jm L pm P p mk L jp [jm] = [jm] + L p jm;p L p jp;m + L p mp;j + L p jm = [jm] + (L p jm;p + Pp pql q P q jm jp Lp qm + (L p mp;j + Pp qj Lq mp P q mj Lp qp = [jm] + L p jm ;p Lp jp ;m + Lp mp P q pj Lp mq ;j + Lp jm P q mpl p jq ) + L p L q jm pq Now accordng to (.7) (.0) and (.8) one obtans ) (L p jp;k + Pp qml q P q jp jm Lp qp + (P p pql q jm P q jp Lp qm P q mpl p ) jq P q pml p ) jq P p pql q jm P q jp Lp qm P q mpl p = (δ p jq qψ p + δ p pψ q )L q (δ q jm j ψ p + δ q pψ j )L p qm (δ q mψ p + δ q pψ m )L p jq = L p jm ψ p + NL p jm = (N )L p jm ψ p L p jm ψ p + L p mp ψ p L p pm ψ j L q jq ψ m = 0 ψ j L p jm ψ p L q ψ jq m Therefore [jm] = [jm] + L p jm;p Lp jp;m + Lp mp;j + Lp L q jm pq. It s easy to see that [jm] = [jm] [jm] = [jm] holds. On the same way we get the other equvalences. Immedately follows Corollary.. Let f : GA N GA N be equtorson geodesc mappng of two equaffne spaces of the -knd {0... } whch satsfes the condton (.8) then [jm] [jm] [jm] [jm] [jm] [jm] are nvarants of ths mappng.

8 M.S. Stankovć et al. / Flomat 6: (0) 9 6 Specally n the generalzed emannan space G N t s vald that Γ p p tensor s always symmetrc so the equatons (.) are reduced to [jm] = Γ p jm;p [jm] = Γ p jm;p [jm] = Γ p jm;p [jm] = Γ p jm;p [jm] = 0. Corollary.. In the space G N s vald [jm] = [jm] = [jm] = [jm] [jm] = 0.e. the ffth cc tensor jm s always symmetrc. = Γ p p = 0 (see []) and cc Also ψ s always gradent and f the spaces G N and G N are equaffne of the -knd { } then the other cc tensors are symmetrc too. The equalty (.) holds and t s reduced to Γ α ψ mj α = 0. The smplfed connectons between curvature tensors n the case of equtorson geodesc mappng of the generalzed emannan spaces whch are equaffne of -knd {0... } are: jmn = jmn = jmn = jmn = jmn = jmn + δ mψ jmn + δ mψ jmn + δ mψ jmn + δ mψ jmn + δ mψ jn δ nψjm + L mnψ j jn δ nψjm + L nmψ j jn δ nψ jn δ nψ jn δ nψjm jm + L mj jm + L mj where ψjm = ψ j ψ j ψ m α {... }. α αm ψ n + L nj ψ n + L nj ψ m ψ m. Equtorson geodesc mappngs of ant-equaffne spaces Defnton.. The general affne connecton space GA N s ant-equaffne of the -knd {... } f the cc tensor of the -knd s ant-symmetrc.e jm = mj. The space GA N s zero-ant-equaffne f the cc tensor s ant-symmetrc.e. j = j. In the sequel we wll descrbe equtorson geodesc mappngs of two ant-equaffne spaces of the -knd {0.. }. By symmetrzaton wth respect to the ndces j and m wthout dvson n (.) we get (jm) = (jm) L p jp (jm) = (jm) + L p jp (jm) = (jm) + L p jp (jm) = (jm) + L p jp (jm) = (jm) + L p jq ;m Lp mp ;m + Lp mp ;m + Lp mp ;m + Lp mp L q pm. ;j Lp jq ;j Lp jq ;j Lp jq ;j + 6Lp jq L q pm L q pm L q pm L q pm where ( ) denotes symmetrzaton wthout dvson. (.)

9 emark.. In the space GA N s vald (jm) = (jm). M.S. Stankovć et al. / Flomat 6: (0) 9 7 Let us denote ψ jm = ψ j;m ψ j ψ m where ψ j s gven at (.8) and (;) s covarant dfferentaton wth respect to the symmetrc part of connecton L. Theorem.. Let f : GA N GA N be equtorson geodesc mappng of two ant-equaffne spaces of -knd {0 } then ψ j s ant-symmetrc. Proof. By symmetrzaton of the equaton (.7) wth respect to the ndces j and m we get where (jm) = (jm) + N ( ψ (jm) + ψ(jm)) ψ Fnally (jm) + ψ(jm) = ψ jm L p mj ψ p + ψ mj L p jm ψ p + ψ jm L p jm ψ p + ψ mj L p mj ψ p ψ m ψ j = (ψ jm L p jm ψ p + ψ mj L p mj ψ p ψ m ψ j ) = (ψ jm + ψ mj ) = ψ (jm). (jm) = (jm) + ( N)ψ (jm). (.) Suppose that the spaces GA N and GA N are ant-equaffne of the ffth knd. Then ψ (jm) = 0.e. ψ jm s antsymmetrc. Otherwse usng (.) and the fact that the correspondng torsons are equal under equtorson mappng we have (jm) = (jm) + ( N)ψ (jm). (.) Suppose that GA N and GA N are ant-equaffne spaces of the zero knd then ψ (jm) = 0 and ψ jm s antsymmetrc. Obvously under equtorson geodesc mappng of two ant-equaffne spaces of -knd {0 } ψ j = 0 f and only f ψ s gradent. From the equatons (.) and (.) we have Corollary.. Let f : GA N GA N be equtorson geodesc mappng. A necessary and suffcent that the tensor (jm) s nvarant of ths mappng s that the tensor (jm) s nvarant of ths mappng. Corollary.. If f : GA N GA N s equtorson geodesc mappng of two ant-equaffne spaces of the zero knd then the tensor (jm) s an nvarant of ths mappng. And nversely f f : GA N GA N s equtorson geodesc mappng of two ant-equaffne spaces of the ffth knd then the tensor (jm) s an nvarant of ths mappng. Theorem.. Let f : GA N GA N be equtorson geodesc mappng of two ant-equaffne spaces of the -knd {... }. The magntude ψ j s antsymmetrc f and only f L α mα ψ j + L α jα ψ m = 0. (.)

10 M.S. Stankovć et al. / Flomat 6: (0) 9 8 Proof. Assume that the spaces GA N and GA N are ant-equaffne of the frst knd. The relaton between the frst curvature tensors of the spaces GA N and GA N under equtorson geodesc mappng s jmn = jmn + δ j (ψ mn ψ nm) + δ mψ jn δ nψjm + L mnψ j + L α mnψ α δ j. (.) By symmetrzaton wth respect to the ndces j and m wthout dvson n (.) we get (jm) = (jm) + ( N)ψ(jm) + L α mαψ j + L α jαψ m. As (jm) = 0 (jm) = 0 one obtans (N )ψ(jm) = L α mαψ j + L α ψ jα m. On the other hand ψ(jm) = ψ jm L p jm ψ p + ψ mj L p mj ψ p ψ m ψ j = ψ jm L p jm ψ p + ψ mj L p mj ψ p ψ m ψ j = ψ jm + ψ mj = ψ (jm). Obvously ψ j s ant-symmetrc f and only f the condton (.) s satsfed. On the same way f we start from jm { } we get the same condton. The Corollares. and. don t hold for the other cc tensors n general case but they do n the case of specal equtorson geodesc mappngs. Namely Theorem.. Let f : GA N GA N be equtorson geodesc mappng of two general affne connecton spaces whch satsfes L α mα ψ j + L α jα ψ m = 0. (.6) Then the next condtons are equvalent: a) (jm) = (jm) b) (jm) = (jm) c) (jm) = (jm) d) (jm) = (jm) e) (jm) = (jm) f ) (jm) = (jm). Proof. Let us prove the equvalence (jm) = (jm) (jm) = (jm). Accordng to (.) (.6) and (.) we have (jm) = (jm) L p jp;m L p mp where P q jm Lp qp ;j L p jq = (δ q j ψ m + δ q mψ j )L p qp L q pm = (jm) L p jp ;m Lp mp ;j Lp jq L q pm + P q jm Lp qp = (ψ m L p + ψ jp j L p mp) = 0.e. (jm) = (jm) L p jp;m Lp mp;j Lp jq easy to see that (jm) = (jm) (jm) = (jm) holds. On the same way we get the other equvalences. L q pm. It s Immedately follows Corollary.. Let f : GA N GA N be equtorson geodesc mappng of two ant-equaffne spaces of -knd {0... } whch satsfes the condton (.6) then (jm) (jm) (jm) (jm) (jm) (jm) are nvarants of ths mappng.

11 M.S. Stankovć et al. / Flomat 6: (0) 9 9 Specally n the generalzed emannan space G N (G N ) the equatons (.) reduce to (jm) = jm Γ p jq (jm) = jm Γ p jq (jm) = jm Γ p jq (jm) = jm + 6Γ p jq jm = jm + Γ p jq Γ q pm Γ q pm Γ q pm Γ q pm Γ q pm (jm) = jm Γ p jq (jm) = jm Γ p jq (jm) = jm Γ p jq (jm) = jm + 6Γ p jq jm = jm + Γ p jq Γ q pm Γ q pm Γ q pm Γ q pm Γ q pm Corollary.. In the space G N s vald (jm) = (jm) = (jm).. Corollary.. Under equtorson geodesc mappng of two generalzed emannan spaces G N and G N the next tensors are nvarants of ths mappng: (jm) jm (jm) jm (jm) jm (jm) jm jm jm. Let f : G N G N be equtorson geodesc mappng of two generalzed emannan spaces G N and G N. In ths case vector ψ j s always gradent and jm jm jm jm are symmetrc. The equaton (.) always holds. Suppose that some other par of cc tensors s ant-symmetrc for example jm and jm. Then we have ψ (jm) = 0 and accordng to Corollary. jm and jm are nvarants of that mappng. As ψ j s gradent we get 0 = ψ jm = ψ jm L p jm ψ p ψ j ψ m + L p jm ψ jm = ψjm = L p mj the curvature tensors jmn = jmn = jmn = jmn = ψ p = ψjm + L p ψ jm p.e. ψjm = ψjm = L p ψ jm p and on the same way ψ p. Usng these facts and (.6) (.7) (.0) (.) and (.) we get the connectons between jmn δ mγ p jn jmn δ mγ p nj ψ p + δ nγ p ψ jm p + Γ mnψ j ψ p + δ nγ p ψ mj p + Γ nmψ j jmn δ j Γp nm ψ p δ mγ p nj jmn δ j Γp nm ψ p δ mγ p nj jmn = jmn jmn = jmn. ψ p δ nγ p mj ψ p δ nγ p mj ψ p + Γ mj ψ p + Γ mj ψ n + Γ nj ψ n + Γ nj If we start from the tensors jm and jm { } and suppose ther ant-symmetry we get the same equatons (.7). Obvously Theorem.. Let f : G N G N be equtorson geodesc mappng of two generalzed emannan spaces G N and G N whch are ant-equaffne of -knd {... } then the tensors jmn + jmn jmn jmn jmn jmn are the nvarants of that mappng. In [7] several cc type denttes are obtaned by usng non-symmetrc affne connecton. In these denttes curvature tensors... 8 appear. In (.) fve ndependent tensors are gven and the rest can be expressed as ther lnear combnatons and of the tensor gven at (.). Followng the notaton n [7] we get ψ m ψ m (.7)

12 M.S. Stankovć et al. / Flomat 6: (0) 9 0 Corollary.6. Under equtorson geodesc mappng of two ant-equaffne generalzed emannan spaces of -knd {... } the next curvature tensors are nvarant: jmn jmn = jmn ( jmn + jmn ) jmn = jmn jmn = jmn jmn.. Concluson It s clearly that a curve γ s a geodesc of connecton L f and only f t s a geodesc of the symmetrzed connecton L = (L + L kj ). Any nvarant object of the projectve class of the connecton L s also nvarant object of the projectve class of the connecton L but t s not vald nversely. Snce the tensors {... } are generalzatons of emannan curvature tensor then the jmn magntudes E {... } are generalzatons of the Weyl projectve curvature tensor [9]. That means than n geodesc mappng for fndng some others nvarants the antsymmetrc part of connecton would be ncluded (.7). Ths fact means that geodesc mappng of two non-symmetrc affne connecton spaces has a sense. In the case of symmetrc affne connecton spaces cc tensors of the -knd {... } reduce to the cc tensor obtaned from the emannan curvature tensor (.). Geodesc mappng of two equaffne spaces of the -knd reduces to the geodesc mappng of two equaffne spaces. Under such mappng vector ψ s always gradent. Ant-equaffne spaces of the -knd {... } reduce to the ant-equaffne spaces of symmetrc connecton when cc tensor s ant-symmetrc. Specally generalzed emannan spaces are nterestng n whch cc tensor jm and cc tensor of the ffth knd jm are always symmetrc. Under equtorson geodesc mappng of two ant-equaffne generalzed emannan spaces of the -knd {... } the curvature tensors jmn and are nvarants. jmn eferences [] A. Ensten A Generalzaton of the elatvstc Theory of Gravtaton Annals of Mathematcs 6() (9) [] A. Ensten elatvstc Theory of the Non-Symmetrc Feld Appendx II n The Meanng of elatvty Ffth Edton Prnceton 9. [] L. P. Esenhart Generalzed emannan spaces I Proceedng of the Natonal Academy of Scences of the USA 7() (9). [] G. S. Hall D. P. Lone The prncple of equvalence and projectve structure n spacetmes Classcal and Quantum Gravty () (007) [] G. S. Hall D. P. Lone The prncple of equvalence and cosmologcal metrcs Journal of Mathematcal Physcs 9() (008) 00. [6] G. S. Hall D. P. Lone Projectve equvalence of Ensten spaces n general relatvty Classcal and Quantum Gravty 6() (009) 009. [7] J. Mkeš Geodesc mappngs of specal emannan spaces Colloqua Mathematca Socetats János Bolya Topcs n Dfferental Geometry Debrecen (Hungary) 6 (98) [8] J. Mkeš Geodesc mappngs of Ensten spaces Mathematcal Notes 8(6) (98) 9 9; Matematcheske Zametk 8(6) (980) [9] J. Mkeš Geodesc mappngs of affne-connected and emannan spaces Journal of Mathematcal Scences 78() (996). [0] J. Mkeš V. Berezovsk Geodesc mappngs of affne-connecton spaces nto emannan spaces Colloqua Mathematca Socetats János Bolya Dfferental Geometry Eger (Hungary) 6 (989) 9 9. [] J. Mkeš I. Hnterletner V. Kosak On geodesc mappngs of affne connecton manfolds Acta Physca Debrecna (008) 9 8. [] J. Mkeš V. Kosak A. Vanžurová Geodesc Mappngs of Manfolds wth Affne Connecton Olomouc 008. [] J. Mkeš A. Vanžurová I. Hnterletner Geodesc Mappngs and Some Generalzatons Olomouc 009. [] J. Mkeš I. Hnterletner On geodesc mappngs of manfolds wth affne connecton arxv:090:89v[math.dg] (009). [] S. M. Mnčć cc denttes n the space of non-symmetrc affne connecton Matematčk Vesnk 0() (97) 6 7. [6] S. M. Mnčć New commutaton formulas n the non-symmetrc affne connecton space Publcatons de L Insttut Mathématque (6) (977)

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