Appled Mathematcs Letters (0) 665 67 Contents lsts avalable at ScenceDrect Appled Mathematcs Letters journal homepage: www.elsever.com/locate/aml On equtorson geodesc mappngs of general affne connecton spaces onto generalzed Remannan spaces Mlan Lj. Zlatanovć Faculty of Scences and Mathematcs, Unversty of š, Všegradska, 8000 š, Serba a r t c l e n f o a b s t r a c t Artcle hstory: Receved 5 January 00 Receved n revsed form 9 ovember 00 Accepted December 00 Keywords: Geodesc mappng General affne connecton space Generalzed Remannan space Equtorson geodesc mappng In the papers Mnčć (97) [5], Mnčć (977) [6], several Rcc type denttes are obtaned by usng non-symmetrc affne connecton. Four knds of covarant dervatves appear n these denttes. In the present work, we consder equtorson geodesc mappngs f of two spaces GA and GR, where GR has a non-symmetrc metrc tensor,.e. we study the case when GA and GR have the same torson tensors at correspondng ponts. Such a mappng s called an equtorson mappng Mnčć (997) [], Stankovć et al. (00) [], Stankovć (n press) []. The exstence of a mappng of such type mples the exstence of a soluton of the fundamental equatons. We fnd several forms of these fundamental equatons. Among these forms a partcularly mportant form s system of partal dfferental equatons of Cauchy type. 00 Elsever Ltd. All rghts reserved.. Introducton A generalzed Remannan space GR s a dfferentable -dmensonal manfold, equpped wth non-symmetrc metrc tensor g j. Generalzed Crstoffel s symbols of the frst knd of the space GR are gven by the formula Γ.jk = (g j,k jk, + g k,j ), where, for example, g j,k = g j / x k. Connecton coeffcents of the space GR are the generalzed Crstoffel s symbols of the second knd Γ = jk g p Γ p.jk, where (g j ) = (g j ) and j denotes a symmetrzaton wth dvson wth respect to the ndces and j. Generally we have Γ Γ jk kj. We suppose that g = det(g j) 0, g = det(g j ) 0. A general affne connecton space GA s a dfferentable -dmensonal manfold, wth non-symmetrc connecton coeffcents L jk. Geodesc mappngs and ther generalzatons were nvestgated by many authors, for example: Snyukov [], Mkeš [ 6], Kosak [5], Vanžurová [5,6], Berezovsk [], Hnterletner [6], Hall and Lone [7 9], Prvanovć [0], Mnčć [ ], Stankovć [ ] and many others. Many authors asked f t makes sense to consder geodesc mappngs between two spaces wth non-symmetrc connectons whereas the defnton of geodescs ncludes only symmetrc connectons. In [], Mnčć and Stankovć showed that t s possble. Ths fact enables further consderaton of geodesc mappngs when the connecton s non-symmetrc (see [ ]). Let us consder two -dmensonal manfolds GA and GR and dfferentable mappng f : GA GR. We can consder these manfolds n the common system of local coordnates wth respect to ths mappng (see Fg..). amely, f f : M GA M GR and f (U, ϕ) s local chart around the pont M, t wll be ϕ(m) = x = (x,..., x ) E-mal address: zlatmlan@yahoo.com. 089-9659/$ see front matter 00 Elsever Ltd. All rghts reserved. do:0.06/j.aml.00..00
666 M.Lj. Zlatanovć / Appled Mathematcs Letters (0) 665 67 Fg... Manfolds n the common system of local coordnates. E (Eucldean -space). In ths case, we defne for the coordnate mappng n the GR the mappng ϕ = ϕ f, and then ϕ(m) = (ϕ f ) (f (M)) = ϕ(m) = x = (x,..., x ) E, (.) wherefore the ponts M and M = f (M) have the same local coordnates. A geodesc mappng [ 5,,] of GA onto GR s a dffeomorphsm f : GA GR under whch the geodescs of the space GA correspond to the geodescs of the space GR. At the correspondng ponts M and M, we can put Γ jk = L jk + P jk, (, j, k =,..., ), where P jk s the deformaton tensor of the connecton L jk of GA accordng to the mappng f : GA GR. The tensor P jk s non-symmetrc wth respect to the ndces j and k, because L jk and Γ jk are non-symmetrc. A necessary and suffcent condton for the mappng f to be geodesc [] s that the deformaton tensor P jk from (.) has the form where P jk = δ j ψ k + δ k ψ j + ξ jk, ψ = + (Γ L ), (.) (.) ξ = jk P jk = (P jk P kj ). (.) We remark that n GR the condton below holds true (see []): Γ = Γ [] = 0, (.5) where j denotes the symmetrzaton, j-antsymmetrzaton, [... j] denotes the antsymmetrzaton wthout dvson wth respect to the ndces, j, and also (... j) denotes the symmetrzaton wthout dvson wth respect to the ndces, j. In GA (GR ), one can defne four knds of covarant dervatves [5,6]. For example, for a tensor a j, we have a j a j = m a + j,m L pm ap j L p jm a, p a j = m a + j,m L mp ap j L p mj a, p = m a + j,m L pm ap j L p mj a, p a j = m a + j,m L mp ap j L p jm a. p Remark.. Let GA be an -dmensonal dfferentable manfold, on whch a non-symmetrc affne connecton L jk s ntroduced. Because of the non-symmetry of the connecton L jk, another connecton can be defned by L = jk L kj. Denote by, the covarant dervatve of the knd θ, (θ =,,, ) n GA and GR, respectvely. θ θ Whereas n a Remannan space (the space of General Relatvty Theory), the connecton coeffcents are expressed n terms of the symmetrc metrc tensor g j, n Ensten s work n Unfed Feld Theores (950 955), the relaton between these magntudes s determned by the followng equaton: g j ;m g j,m Γ p m g pj Γ p mj g p = 0, g j,m = g j. (.6) + x m In the Eq. (.6), the ndex behaves n the sense of the frst knd of dervatve ( ), and the ndex j n the sense of the second one ( ).
M.Lj. Zlatanovć / Appled Mathematcs Letters (0) 665 67 667 Ensten n [7], 950, for the covarant curvature tensor n hs theory obtans a Banch-type dentty: R klm + + ;n + R kmn +++ ;l + R knl + ;m = 0, where R klm = g p R p klm, and the ndces behave n the sense as explaned n the comment just below relaton (.6). In the case of the space GA (GR ), we have fve ndependent curvature tensors [8,9] (n [8] R 5 s denoted by R ): (.7) R = jmn L j[m,n] + Lp j[m L pn], R = jmn L [mj,n] + Lp [mj L, n]p R = jmn L jm,n L + nj,m Lp jm L np Lp nj L + pm Lp nm L [pj], R = jmn L jm,n L + nj,m Lp jm L np Lp nj L + pm Lp mn L [pj], R = jmn (L j[m,n] + L [mj,n] + Lp jm L + pn Lp mj L np Lp jn L mp Lp nj L ). pm 5 In a Remannan space, the Eq. (.) s equvalent to Lev-Cvta s equaton (see []): (.8) g j;k = ψ k g j + ψ g jk + ψ j g k, where (; ) s the covarant dervatve n the space R,.e. g j;k connecton. (.9) = g j / x k Γ p k g pj Γ p jk g p, and Γ s the Lev-Cvta s Theorem. ([]). Generalzed Remannan space GR admts nontrval geodesc mappngs onto generalzed Remannan space GR f and only f for the metrc tensor of the space GR s vald: g j k = g j + ψ k kg j + ψ g kj + ψ j g k + ξ p k g pj + ξ p jk g p, (.0) where ( ) and ( ) are covarant dervatves n the spaces GR and GR, respectvely. The condton (.) s equvalent to (.0). It can easly be seen that for the second, thrd and fourth knd of covarant dervatves equatons smlar to (.0) can be derved.. Equtorson geodesc mappngs A geodesc mappng f : GA GR s an equtorson geodesc mappng f the torson tensors of the spaces GA and GR are equal n the common local coordnates. Then from (.) (.), we get Γ h j L h j = ξ h j = 0, where j denotes the antsymmetrzaton wth respect to the ndces, j (see [ ]). Mkeš and Berezovsk proved n [ 5] the followng theorem: (.) Theorem.. The manfold wth affne connecton A admts geodesc mappng onto Remannan manfold R wth the metrc tensor g j f and only f the followng set of dfferental equatons of Cauchy type wth covarant dervatves has a soluton wth respect to the symmetrc tensor: g j, (det(g j ) 0), the covector ψ and the functon µ. (a) g j;k = ψ k g j + ψ g jk + ψ j g k ; (b) ψ ;j = ψ ψ j + µg j βγ g R βγ j R j + R ; j (c) ( )µ ; = ( )ψ g βγ R βγ + ψ g β 5R β + 6 + Rγ γ β R β + g β R γ β;γ R ;β + Rγ γ, (.) where (;) denotes covarant dervatve wth respect to the connecton of A, (g j ) s the matrx nverse to (g j ), R h jk, R j are respectvely Remannan and Rcc tensors of the manfold A, and R j = g βγ R βγ j, R j. β = g βγ R jγ, R ;. β = g βγ R ;γ and R.;β = g βγ R γ ;.
668 M.Lj. Zlatanovć / Appled Mathematcs Letters (0) 665 67 We gve some generalzatons of ths theorem n the case of manfolds wth a non-symmetrc metrc tensor. From (.0) and (.), we have g j k = ψ k g j + ψ g kj + ψ j g k = g j k. Further, we obtan g j ks j sk = g j ψ [ks] + g k( ψ (.) j)s s( ψ j)k, (.) where ψ jk = ψ j k ψ j ψ k. Usng the approprate Rcc dentty [5], from (.), one gets.e. g j ks j sk = g R jks g j R ks Lp [ks] g j p, R jks g j R Transvectng the last equaton by g j, we get where ψ ψ {ks} = + R ks Lp [ks] (ψ pg j + ψ g pj + ψ j g p ) = g j ψ, ks {ks} = ψ [ks] + ψ p L p [ks]. Replacng (.6) n (.5), we obtan ( R + j)ks + g j R Transvectng ths equaton by g jk, we get jk g R + jks R s + + R Usng (.5) and (.), we get where µ ψ g j j = ψ ψ j + µ ks Lp [ks] ψ (g pj) = g k( ψ s Lp [ps] ψ jk g p L p [ks] ψ j = ψ [ks] + g k( ψ j)s s( ψ j)k. (.5) (.6) j)s s( ψ j)k. (.7) s jk g s ψ jk. (.8) g j βγ g R βγ + j R j + + R ψ β j L.[βj], (.9) = g jk ψ jk and ψ j = g j ψ. Because of g k g j = δ k j, one obtans k = ψ kg j δ k ψ j δ j k ψ = g j From (.9), we obtan. k ψ jk ψ kj = ψ kψ j + ψ ψ j k + µ kg j + µ g j k βγ k g R βγ g R βγ j + k R j k + + R j βγ j gβγ g k R βγ j ψ β k k L.[βj] ψ β L.[βj] k ψ jψ k ψ ψ k j µ jg k µ g k j + g βγ j g R + g βγ g R βγ k j R k j + R k βγ + k gβγ g j R βγ k (.0) + ψ β j j L.[βk] + ψ β L.[βk] j. (.) Takng nto account (.), (.9), (.0), contractng wth g j n (.) and usng the correspondng Rcc dentty [5], we get that the left sde of the Eq. (.) s L = g j ψ p R and the rght sde s p jk j L p [jk] ψ p D = ( )g βγ R βγ ψ k + g j R k ψ j + 6 + g j R ψ k j + ( )g β L γ [βk] ψ ψ γ + ( ) µ k ψ γ R γ + k gβγ R βγ k j R k j + g j R k j + gβ L γ [βk] γ ψ + g β L γ [βk] ψ γ. (.)
M.Lj. Zlatanovć / Appled Mathematcs Letters (0) 665 67 669 From L = D, we get ( ) µ k = ( )g βγ R βγ ψ k j R k ψ j 6 + g j R ψ k j ( )g β L γ [βk] ψ ψ γ In GA, (see [0]), the followng s vald: S R = jmn R + jmn R + mnj R jmn and fnally, replacng n (.), we get + ψ γ R γ k gβγ R βγ k + g j R k ( + ) β L γ [βk] ψ ψ γ sq p g p R sqγ + R γ + + R njm = S jmn + g j R k j + j gβ L γ [βk] γ ψ p pγ. (.) (L + [jm],n Lp [jm] L pn ), (.) ( ) µ k = ( )g β p ψ p R ψ βk g 5 β R βk + 6 + R γ γ βk R kβ R β γ βk γ R k β + R γ γ k β ( + ) β L γ [βk] g sq p g p R sqγ + R γ + + R p pγ So, the next theorem s proved. ( )g β L γ [βk] ψ ψ γ β L γ [βk] γ ψ + ψ g β S L p [qβ] Lq. pk (.5) Theorem.. If the manfold wth general affne connecton GA admts equtorson geodesc mappng onto generalzed Remannan manfold GR wth the metrc tensor g j, then the followng set of dfferental equatons wth covarant dervatves of the frst knd of Cauchy type has a soluton wth respect to the symmetrc tensor g j, the covector ψ and the functon µ : (a) g j k = ψ k g j + ψ g kj + ψ j g k ; (b) ψ (c) ( ) µ k j = ψ ψ j + µ g j βγ g R βγ + j R j + + R j gβ g γ L γ [βj] ψ ; = ( )g β p ψ p R ψ βk g 5 β R βk + 6 + R γ γ βk R kβ R β γ βk γ R k β + R γ γ k β ( )g β L γ [βk] ψ ψ γ β L γ [βk] ( + ) β L γ [βk] g sq p g p R sqγ + R γ + + R p pγ + ψ g β S L p [qβ] Lq. pk γ ψ (.6) Followng ths procedure, the next theorems can be proved. Theorem.. If the manfold wth general affne connecton GA admts equtorson geodesc mappng onto generalzed Remannan manfold GR wth the metrc tensor g j, then the followng set of dfferental equatons wth covarant dervatves of the second knd of Cauchy type has a soluton wth respect to the symmetrc tensor g j, the covector ψ and the functon: µ = g jk ψ jk (a) g j k = ψ k g j + ψ g kj + ψ j g k ; (b) ψ (c) ( ) µ k j = ψ ψ j + µ g j βγ g R βγ + j R j + + R + j gβ g γ L γ [βj] ψ ; = ( )g β p ψ p R ψ βk g 5 β R βk + 6 + R γ γ βk R kβ R β γ βk γ R k β + R γ γ k β + ( )g β L γ [βk] ψ ψ γ + g β L γ [βk] ( + ) + g β L γ [βk] g sq p g p R sqγ + R γ + + R p pγ + ψ g β S L p [βq] Lq. kp γ ψ (.7)
670 M.Lj. Zlatanovć / Appled Mathematcs Letters (0) 665 67 Theorem.. If the manfold wth general affne connecton GA admts equtorson geodesc mappng onto generalzed Remannan manfold GR wth the metrc tensor g j, then the followng set of dfferental equatons wth covarant dervatves of the thrd knd of Cauchy type has a soluton wth respect to the symmetrc tensor g j, the covector ψ and the functon µ = g jk ψ jk (a) g j k = ψ k g j + ψ g kj + ψ j g k ; (b) ψ (c) ( ) µ k j = ψ ψ j + µ g j βγ g R βγ + j R j + + R + j gβ g γ L γ [βj] ψ ; = ( )g β p ψ p R ψ βk g 5 β R βk + 6 + R γ γ βk R kβ R β γ βk γ R k β + R γ γ k β + ( )g β L γ [βk] ψ ψ γ + g β L γ [βk] ( + ) + g β L γ [βk] g sq p g p R sqγ + R γ + + R p pγ + ψ g β S L p [βq] Lq. kp γ ψ (.8) Theorem.5. If the manfold wth general affne connecton GA admts equtorson geodesc mappng onto generalzed Remannan manfold GR wth the metrc tensor g j, then the followng set of dfferental equatons wth covarant dervatves of the fourth knd of Cauchy type has a soluton wth respect to the symmetrc tensor g j, the covector ψ and the functon µ = g jk ψ jk (a) g j k = ψ k g j + ψ g kj + ψ j g k ; (b) ψ (c) ( ) µ k j = ψ ψ j + µ g j βγ g R βγ + j R j + + R j gβ g γ L γ [βj] ψ ; = ( )g β p ψ p R ψ βk g 5 β R βk + 6 + R γ γ βk R kβ R β γ βk γ R k β + R γ γ k β ( )g β L γ [βk] ψ ψ γ β L γ [βk] ( + ) β L γ [βk] g sq p g p R sqγ + R γ + + R p pγ + ψ g β S L p [qβ] Lq. pk γ ψ (.9) Systems (.6) (.9) have no more than one soluton for the followng ntal condton at the pont x 0 : g j (x 0 ) = 0 g j, ψ (x 0 ) = ψ 0, µ θ (x 0 ) = µ 0, θ =,,,. General solutons of Eqs. (.6) (.9) depend on a fnte number of substantal parameters ( + ) ( + ) r r 0. θ Fndng all solutons of (.6) (.9) requres consderng ther ntegrablty condtons and dfferental extensons, whch form a set of algebrac equatons wth respect to the unknown functons g j, ψ and µ, θ =,,,, wth coeffcent from θ GA. But ths would certanly be a farly dffcult work to be done.. Concluson We consder equtorson geodesc mappngs [ ] and gve new generalzatons of the mappng f : GA GR. In ths way, we extend some recently obtaned results from [ 6] where geodesc mappngs were nvestgated of an affne connected space onto a Remannan space (n the symmetrc case). As corollares, we get extensons of the correspondng results concernng geodesc mappngs of an affne connected space onto a Remannan space from [ 6] usng a non-symmetrc metrc tensor and the four knds of covarant dervatves. We also use the technques developed n cted papers. We emphasze the followng results of the paper: It s possble to extend the concept of a geodesc mappng of an affne connected space onto a Remannan space, by consderng equtorson geodesc mappngs. In ths way, equtorson geodesc mappngs are avalable for a wder class of metrcs. It s reasonable to expect that these facts wll be a motvaton n some further nvestgatons of geodesc mappngs, and generally for all extensons from the (A ) R nto the (GA ) GR spaces.
M.Lj. Zlatanovć / Appled Mathematcs Letters (0) 665 67 67 In ths paper, we got four systems of PDEs of Cauchy type n GA. Perhaps n future work we can consder solutons of these systems. Acknowledgements The author gratefully acknowledges the support from the research project 70 of the Serban Mnstry of Scence. The author s also grateful to the anonymous referees for valuable comments and suggestons, whch were most helpful n mprovng the manuscrpt. References [].S. Snyukov, Geodesc Mappngs of Remannan Spaces, auka, Moscow, 979 (n Rusan). [] M. Jukl, L. Juklova, J. Mkeš, On generalzed trace decompostons problems, n: Proceedngs of the rd Internatonal Conference Dedcated to 85-th Brthday of Professor Kudrjavcev, 008, pp. 99. [] J. Mkeš, Geodesc mappngs of affne-connected and Remannan spaces, J. Math. Sc. ew York (996). [] J. Mkeš, V. Berezovsk, Geodesc mappngs of afnely connected spaces onto Remannan spaces, Coll. Math. Soc. János Bolya (989) 9 9. [5] J. Mkeš, V. Kosak, A. Vanžurová, Geodesc mappngs of manfolds wth affne connecton, Olomouc, 008. [6] J. Mkeš, A. Vanžurová, I. Hnterletner, Geodesc mappngs and some generalzatons, Olomouc, 009. [7] G.S. Hall, D.P. Lone, The prncple of equvalence and projectve structure n spacetmes, Class. Quantum Grav. (007) 67 66. [8] G.S. Hall, D.P. Lone, The prncple of equvalence and cosmologcal metrcs, J. Math. Phys. 9 (008) 050. [9] G.S. Hall, D.P. Lone, Projectve equvalence of Ensten spaces n general relatvty, Class. Quantum Grav. 6 (009) 5009. [0] M. Prvanovć, π-projectve Curvature Tensors, n: Analles Unv. Mara Cure-Sklodowska, Lubln - Polona, XLI, vol. 6, 986, pp.. [] S.M. Mnčć, M.S. Stankovć, On geodesc mappng of general affne connecton spaces and of generalzed Remannan spaces, Mat. vesnk 9 (997) 7. [] S.M. Mnčć, M.S. Stankovć, Equtorson geodesc mappngs of generalzed Remannan spaces, Publ. Inst. Math. (Beograd) (.S.) 6 (75) (997) 97 0. [] M.S. Stankovć, S.M. Mnčć, S.Lj. Velmrovć, M.Lj. Zlatanovć, On equtorson geodesc mappngs of general affne connecton spaces, Rend. Sem. Mat. Unv. Padova (n press). [] M.S. Stankovć, M.Lj. Zlatanovć, S.Lj. Velmrovć, Equtorson holomorphcally projectve mappngs of generalzed Kähleran space of the frst knd, Czechoslovak Math. J. 60 () (00) 65 65. [5] S.M. Mnčć, Rcc denttes n the space of non-symmetrc affne connecton, Mat. Vesnk 0 (5) (97) 6 7. [6] S.M. Mnčć, ew commutaton formulas n the non-symmetrc affne connecton space, Publ. Inst. Math. (Beograd) (.S.) (6) (977) 89 99. [7] A. Ensten, The Banch denttes n the generalzed theory of gravtaton, Canadan Jour. Math. (950) 0 8. [8] S.M. Mnčć, Independent curvature tensors and pseudotensors of spaces wth non-symmetrc affne connecton, Coll. Math. Soc. János Bolya (989) 50 60. [9] M. Prvanovć, Four curvature tensors of non-symmetrc affne connexon, n: Proceedngs of the Conference 50 Years of Lobachevsk Geometry, Kazan 976, Moscow 997, pp. 99 05 (n Russan). [0] S.M. Mnčć, Symmetry propertes of curvature tensors of the space wth non-symmetrc affne connecton and generalzed Remannan space, Zb. Rad. Flozofskog fak. u šu, Ser. Mat. () (987) 69 78.