Gemetry f parallelzable manflds n the cntext f generalzed Lagrange spaces M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed Abstract. In ths paper, we deal wth a generalzatn f the gemetry f parallelzable manflds, r the abslute parallelsm (AP-) gemetry, n the cntext f generalzed Lagrange spaces. All gemetrc bjects defned n ths gemetry are nt nly functns f the pstnal argument x, but als depend n the drectnal argument y. In ther wrds, nstead f dealng wth gemetrc bjects defned n the manfld M, as n the case f classcal AP-gemetry, we are dealng wth gemetrc bjects n the pullback bundle π 1 (T M) (the pullback f the tangent bundle T M by π : T M M). Many new gemetrc bjects, whch have n cunterpart n the classcal AP-gemetry, emerge n ths mre general cntext. We refer t such a gemetry as generalzed AP-gemetry (GAP-gemetry). In analgy t AP-gemetry, we defne a d-cnnectn n π 1 (T M) havng remarkable prpertes, whch we call the canncal d-cnnectn, n terms f the unque trsn-free Remannan d-cnnectn. In addtn t these tw d-cnnectns, tw mre d-cnnectns are defned, the dual and the symmetrc d-cnnectns. Our space, therefre, admts twelve curvature tensrs (crrespndng t the fur defned d-cnnectns), three f whch vansh dentcally. Smple frmulae fr the nne nn-vanshng curvatures tensrs are btaned, n terms f the trsn tensrs f the canncal d- cnnectn. The dfferent W -tensrs admtted by the space are als calculated. All cntractns f the h- and v-curvature tensrs and the W -tensrs are derved. Secnd rank symmetrc and skew-symmetrc tensrs, whch prve useful n physcal applcatns, are sngled ut. M.S.C. 2000: 53B40, 53A40, 53B50 Key wrds: Parallelzable manfld, Generalzed Lagrange space, AP-gemetry, GAP-gemetry, Canncal d-cnnectn, W -tensr. Balkan Jurnal f Gemetry and Its Applcatns, Vl.13, N.2, 2008, pp. 120-139. c Balkan Scety f Gemeters, Gemetry Balkan Press 2008.
Gemetry f parallelzable manflds 121 1 Intrductn The gemetry f parallelzable manflds r the abslute parallelsm gemetry (APgemetry) ([5], [10], [14], [15]) has many advantages n cmparsn t Remannan gemetry. Unlke Remannan gemetry, whch has ten degrees f freedm (crrespndng t the metrc cmpnents fr n = 4), AP-gemetry has sxteen degrees f freedm (crrespndng t the number f cmpnents f the fur vectr felds defnng the parallelzatn). Ths makes AP-gemetry a ptental canddate fr descrbng physcal phenmena ther than gravty. Mrever, as ppsed t Remannan gemetry, whch admts nly ne symmetrc lnear cnnectn, AP-gemetry admts at least fur natural (bult-n) lnear cnnectns, tw f whch are nn-symmetrc and three f whch have nn-vanshng curvature tensrs. Last, but nt least, asscated wth an AP-space, there s a Remannan structure defned n a natural way. Thus, APgemetry cntans wthn ts gemetrcal structure all the mathematcal machnery f Remannan gemetry. Accrdngly, a cmparsn between the results btaned n the cntext f AP-gemetry and general relatvty, whch s based n Remannan gemetry, can be carred ut. In ths paper, we study AP-gemetry n the wder cntext f a generalzed Lagrange space ([7], [9], [11], [12]). All gemetrc bjects defned n ths space are nt nly functns f the pstnal argument x, but als depend n the drectnal argument y. In ther wrds, nstead f dealng wth gemetrc bjects defned n the manfld M, as n the case f classcal AP-space, we are dealng wth gemetrc bjects n the pullback bundle π 1 (T M) (the pullback f the tangent bundle T M by the prjectn π : T M M) [1]. Many new gemetrc bjects, whch have n cunterpart n the classcal AP-space, emerge n ths mre general cntext. We refer t such a space as a d-parallelzable manfld r a generalzed abslute parallelsm space (GAP-space). The paper s rganzed n the fllwng manner. In sectn 2, fllwng the ntrductn, we gve a bref accunt f the basc cncepts and defntns that wll be needed n the sequel, ntrducng the ntn f a nn-lnear cnnectn Nµ α. In sectn 3, we cnsder an n-dmensnal d-parallelzable manfld M ([2], [11]) n whch we defne a metrc n terms f the n ndependent π-vectr felds λ defnng the parallelzatn n π 1 (T M). Thus, ur parallelzable manfld becmes a generalzed Lagrange space, whch s a generalzatn f the classcal AP-space. We then defne the canncal d-cnnectn D, relatve t whch the h- and v-cvarant dervatves f the vectr felds λ vansh. We end ths sectn wth a cmparsn between the classcal AP-space and the GAP-space. In sectn 4, cmmutatn frmulae are recalled and sme denttes btaned. We then ntrduce, n analgy t the AP-space, tw ther d-cnnectns: the dual d-cnnectn and the symmetrc d-cnnectn. The nne nnvanshng curvature tensrs, crrespndng t the dual, symmetrc and Remannan d-cnnectns are then calculated, expressed n terms f the trsn tensrs f the canncal d-cnnectn. In sectn 5, a summary f the fundamental symmetrc and skew symmetrc secnd rank tensrs s gven, tgether wth the symmetrc secnd rank tensrs f zer trace. In sectn 6, all pssble cntractns f the h- and v- curvature tensrs are btaned and the cntracted curvature tensrs are expressed n terms f the fundamental tensrs gven n sectn 5. In sectn 7, we study the dfferent W -tensrs crrespndng t the dfferent d-cnnectns defned n the space, agan expressed n terms f the trsn tensrs f the canncal d-cnnectn. Cntractns
122 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed f the dfferent W -tensrs and the relatns between them are then derved. Fnally, we end ths paper by sme cncludng remarks. 2 Fundamental prelmnares Let M be a dfferental manfld f dmensn n f class C. Let π : T M M be ts tangent bundle. If (U, x µ ) s a lcal chart n M, then (π 1 (U), (x µ, y µ )) s the crrespndng lcal chart n T M. The crdnate transfrmatn law n T M s gven by: x µ = x µ (x ν ), y µ = p µ ν y ν, where p µ ν = xµ x and det(p µ ν ν ) 0. Defntn 2.1. A nn-lnear cnnectn N n T M s a system f n 2 functns Nβ α(x, y) defned n every lcal chart π 1 (U) f TM whch have the transfrmatn law (2.1) N α β where p ɛ β σ = pɛ β x σ = 2 x ɛ x β x σ. = pα α p β β N α β + p α ɛ p ɛ β σ yσ, The nn-lnear cnnectn N leads t the drect sum decmpstn T u (T M) = H u (T M) V u (T M), u T M = T M \ {0}, where H u (T M) s the hrzntal space at u asscated wth N supplementary t the vertcal space V u (T M). If δ µ := µ Nµ α α, where µ := x, µ µ := y, then ( µ µ ) s the natural bass f V u (T M) and (δ µ ) s the natural bass f H u (T M) adapted t N. Defntn 2.2. A dstngushed cnnectn (d-cnnectn) n M s a trplet D = (N α µ, Γ α µν, C α µν), where N α µ (x, y) s a nn-lnear cnnectn n T M and Γ α µν(x, y) and C α µν(x, y) transfrm accrdng t the fllwng laws: (2.2) Γ α µ ν = pα α p µ µ pν ν Γα µν + p α ɛ p ɛ µ ν, (2.3) C α µ ν = pα α p µ µ pν ν Cα µν. In ther wrds, Γ α µν transfrm as the ceffcents f a lnear cnnectn, whereas C α µν transfrm as the cmpnents f a tensr. Defntn 2.3. The hrzntal (h-) and vertcal (v-) cvarant dervatves wth respect t the d-cnnectn D (f a tensr feld A α µ) are defned respectvely by: (2.4) A α µ ν := δ νa α µ + A ɛ µγ α ɛν A α ɛ Γ ɛ µν; (2.5) A α µ ν := ν A α µ + A ɛ µc α ɛν A α ɛ C ɛ µν.
Gemetry f parallelzable manflds 123 Defntn 2.4. A symmetrc and nn-degenerate tensr feld g µν (x, y) f type (0, 2) s called a generalzed Lagrange metrc n the manfld M. The par (M, g) s called a generalzed Lagrange space. Defntn 2.5. Let (M, g) be a generalzed Lagrange space equpped wth a nnlnear cnnectn N α µ. Then a d -cnnectn D = (N α µ, Γ α µ,ν, C α µν) s sad t be metrcal wth respect t g f (2.6) g µν α = 0, g µν α = 0. The fllwng remarkable result was prved by R. Mrn [8]. It guarantees the exstence f a unque trsn-free metrcal d-cnnectn n any generalzed Lagrange space equpped wth a nn-lnear cnnectn. Mre precsely: Therem 2.6. Let (M, g) be a generalzed Lagrange space. Let N α µ be a gven nnlnear cnnectn n T M. Then there exsts a unque metrcal d-cnnectn D = (Nµ α, Γ α µν, C α µν) such that Λ α µν := Γ α µν Γ α νµ = 0 and T α µν := C α µν C α νµ = 0. Ths d-cnnectn s gven by Nµ α and the generalzed Chrstffel symbls: (2.7) Γ α µν = 1 2 gαɛ (δ µ g νɛ + δ ν g µɛ δ ɛ g µν ), (2.8) C α µν = 1 2 gαɛ ( µ g νɛ + ν g µɛ ɛ g µν ). Ths cnnectn wll be referred t as the Remannan d-cnnectn. 3 d-parallelzable manflds (GAP-spaces) The Remannan d-cnnectn mentned n Therem 2.6 plays the key rle n ur generalzatn f the AP-space, whch, as wll be revealed, appears natural. Hwever, t s t be nted that the clse resemblance f the tw spaces s deceptve; as they are smlar n frm. Hwever, the extra degrees f freedm n the generalzed AP-space makes t rcher n cntent and dfferent n ts gemetrc structure (see Remark 3.6). We start wth the cncept f d-parallelzable manflds. Defntn 3.1. An n-dmensnal manfld M s called d-parallelzable, r generalzed abslute parallelsm space (GAP-space), f the pull-back bundle π 1 (T M) admts n glbal lnearly ndependent sectns (π-vectr felds) λ(x, y), = 1,..., n. If λ = ( λ α ), α = 1,..., n, then (3.1) λ α λ β = δβ α, λ α λ α = δ j, j where ( λ α ) dentes the nverse f the matrx ( λ α ). Ensten summatn cnventn s appled n bth Latn (mesh) ndces and Greek (wrld) ndces, where all Latn ndces are wrtten n a lwer pstn. In the sequel, we wll smply use the symbl λ (wthut a mesh ndex) t dente any ne f the vectr felds λ ( = 1,..., n) and n mst cases, when mesh ndces appear they wll be n pars, meanng summatn.
124 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed We shall ften use the expressn GAP-space (resp. GAP-gemetry) nstead f d-parallelzable manfld (resp. gemetry f d-parallelzable manflds) fr ts typgraphcal smplcty. Therem 3.2. A GAP-space s a generalzed Lagrange space. In fact, the cvarant tensr feld g µν (x, y) f rder 2 gven by (3.2) g µν (x, y) := λ µ λ ν, defnes a metrc n the pull-back bundle π 1 (T M) wth nverse gven by (3.3) g µν (x, y) = λ µ λ ν Assume that M s a GAP-space equpped wth a nn-lnear cnnectn N α µ. By Therem 2.6, there exsts n (M, g) a unque trsn-free metrcal d-cnnectn D = (N α µ, Γ α µν, C α µν) (the Remannan d-cnnectn). We defne anther d-cnnectn D = (N α µ, Γ α µν, C α µν) n terms f D by: (3.4) Γ α µν := Γ α µν + λ α λ µ ν, (3.5) Cµν α := C α µν + λ α λ µ ν. Here, and dente the h- and v-cvarant dervatves wth respect t the Remannan d-cnnectn D. If and dente the h- and v-cvarant dervatves wth respect t the d-cnnectn D, then (3.6) λ α µ = 0, λ α µ = 0. Ths can be shwn as fllws: λ α µ = δ µ λ α + λ ɛ Γ α ɛµ = δ µ λ α + λ ɛ ( Γ α ɛµ + (δ µ λ α + λ ɛ Γ α ɛµ) λ α j µ ( λ ɛ j λ α µ = 0. Hence, we btan the fllwng λ α λ ɛ j j µ ) = λ ɛ ) = 0. In exactly the same way, t can be shwn that Therem 3.3. Let (M, λ(x, y)) be a GAP-space equpped wth a nn-lnear cnnectn Nµ α. There exsts a unque d-cnnectn D = (Nµ α, Γ α µν, Cµν), α such that λ α µ = λ α µ = 0. Ths cnnectn s gven by Nβ α, (3.4) and (3.5). Cnsequently, D s metrcal: g µν σ g µν σ = 0. Ths cnnectn wll be referred t as the canncal d-cnnectn. It s t be nted that relatns (3.6) are n accrdance wth the classcal APgemetry n whch the cvarant dervatve f the vectr felds λ wth respect t the canncal cnnectn Γ α µν = λ α ( ν λ µ ) vanshes [15]. Therem 3.4. Let (M, λ(x, y)) be a d-parallelzable manfld equpped wth a nnlnear cnnectn Nµ α. The canncal d-cnnectn D = (Nµ α, Γ α µν, Cµν) α s explctly expressed n terms f λ n the frm (3.7) Γ α µν = λ α (δ ν λ µ ), Cµν α = λ α ( ν λ µ ).
Gemetry f parallelzable manflds 125 Prf. Snce λ α ν = 0, we have δ ν λ α = λ ɛ Γ α ɛν. Multplyng bth sdes by λ µ, takng nt accunt the fact that λ µ = δµ, α we get Γ α µν = λ µ (δ ν λ α ) = λ α (δ ν λ µ ). The λ α prf f the secnd relatn s exactly smlar and we mt t. It s t be nted that the cmpnents f the canncal d-cnnectn are smlar n frm t the cmpnents f the canncal cnnectn n the classcal AP-cntext [15], ntng that ν s replaced by δ ν (fr the h-cunterpart) and by ν (fr the v-cunterpart) respectvely (See Table 1). The abve expressns fr the canncal cnnectn seem therefre lke a natural generalzatn f the classcal AP case. By (3.4) and (3.5), n vew f the abve therem, we have the fllwng Crllary 3.5. The Remannan d-cnnectn D = (Nµ α, expressed n terms f λ n the frm (3.8) Γ α µν = C α µν = Γ α µν, C α µν) s explctly λ α (δ ν λ µ λ µ ν ), λ α ( ν λ µ λ µ ν ). A. s a result f the dependence f λ n the velcty vectr y, the n 3 functns λ α ( ν λ µ ), as ppsed t the classcal AP-space, d nt transfrm as the ceffcents f a lnear cnnectn, but transfrm accrdng t the rule (3.9) λ α ( ν λ µ ) = p α α p µ µ pν ν λ α ( ν λ µ ) + p α ɛ p ɛ µ ν + pα α p µ µ pν ν ɛ Cµν. α yɛ Smlarly, t can be shwn that, n general, tensrs n the cntext f the classcal AP-space d nt transfrm lke tensrs n the wder cntext f the GAP-space; ther dependence n the velcty vectr y spls ther tensr character. In ther wrds, tensrs n the classcal AP-cntext d nt necessarly behave lke tensrs when they are regarded as functns f pstn x and velcty vectr y. Ths means that thugh the classcal AP-space and the GAP-space appear smlar n frm, they dffer radcally n ther gemetrc structures. We nw ntrduce sme tensrs that wll prve useful later n. Let (3.10) γµν α := λ α λ µ ν = Γ α µν Γ α µν, G α µν := λ α λ µ ν = Cµν α C α µν. In analgy t the AP-space, we refer t γµν α and G α µν as the h- and v-cntrtn tensrs respectvely. Let (3.11) Λ α µν := Γ α µν Γ α νµ = γµν α γνµ. α be the trsn tensr f the canncal cnnectn Γ α µν and (3.12) Ω α µν := γµν α + γνµ. α Smlarly, let (3.13) Tµν α := Cµν α Cνµ α = G α µν G α νµ be what we may call the trsn tensr f Cµν α and (3.14) Dµν α := G α µν + G α νµ. Nw, f γ σµν := g ɛσ γµν ɛ and G σµν := g ɛσ G ɛ µν, then γ σµν and G σµν are skew symmetrc n the frst par f ndces. Ths, n turn, mples that (3.15) γɛν ɛ = G ɛ ɛν = 0. Hence, f β µ := γµɛ, ɛ B µ := G ɛ µɛ, then (3.16) Λ ɛ µɛ = γµɛ ɛ = β µ, Tµɛ ɛ = G ɛ µɛ = B µ.
126 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed Table 1. Cmparsn between the classcal AP-gemetry and the GAP-gemetry Classcal AP-gemetry GAP-gemetry Buldng blcks λ α (x) λ α (x, y) Metrc gµν (x) = λµ(x) λν (x) gµν (x, y) = λµ(x, y) λν (x, y) Remannan cnnectn Γ α µν = 1 2 gαɛ { µ g νɛ + ν g µɛ + ɛ g µν } Γ α µν = 1 2 gαɛ {δ µ g νɛ + δ ν g µɛ + δ ɛ g µν } C α µν = 1 2 gαɛ { µg νɛ + νg µɛ + ɛg µν} Canncal cnnectn Γ α µν = λ α ( ν λµ) Γ α µν = λ α (δν λµ) (h-cunterpart) Cµν α = λ α ( ν λµ) (v-cunterpart) AP-cndtn λ α µ = 0 λα µ = 0 (h-cvarant dervatve) λ α µ = 0 (v-cvarant dervatve) Trsn Λ α µν = Γα µν Γα νµ Λ α µν = Γα µν Γα νµ (h-cunterpart) T α µν = Cα µν Cα νµ (v-cunterpart) Cntrsn γ α µν = Γα µν Γ α µν γ µν α = Γα µν Γ α νµ (h-cunterpart) G α µν = Cα µν C α µν (v-cunterpart) Basc vectr βµ = Λ α µα = γα µα βµ = Λ α µα = γα µα (h-cunterpart) Bµ = T α µα = Gα µα (v-cunterpart) Fnally, t can be shwn, n analgy t the classcal AP-space [3], that the cntrtn tensrs γ µνσ and G µνσ can be expressed n terms f the trsn tensrs n the frm (3.17) γ µνσ = 1 2 (Λ µνσ + Λ σνµ + Λ νσµ ) (3.18) G µνσ = 1 2 (T µνσ + T σνµ + T νσµ ), where Λ µνσ := g ɛµ Λ ɛ νσ and T µνσ := g ɛµ T ɛ νσ. It s clear by (3.11), (3.13), (3.17) and (3.18) that the trsn tensrs vansh f and nly f the cntrtn tensrs vansh. Table 1 gves a cmparsn between the fundamental gemetrc bjects n the classcal AP-gemetry and the GAP-gemetry. Smlar bjects f the tw spaces wll be dented by the same symbl. As prevusly mentned, h stands fr hrzntal whereas v stands fr vertcal. 4 Curvature tensrs n Generalzed AP-space Owng t the exstence f tw types f cvarant dervatves wth respect t the canncal cnnectn D, we have essentally three cmmutatn frmulae and cnsequently three curvature tensrs.
Gemetry f parallelzable manflds 127 Lemma 4.1. Let [δ σ, δ µ ] := δ σ δ µ δ µ δ σ and let [δ σ, µ ] be smlarly defned. Then (4.1) [δ σ, δ µ ] = R ɛ σµ ɛ, [δ σ, µ ] = ( µ N ɛ σ) ɛ, where R α σµ := δ µ N α σ δ σ N α µ s the curvature tensr f the nn-lnear cnnectn N α µ. Therem 4.2. The three cmmutatn frmulae f λ α crrespndng t the canncal cnnectn D = (Nµ α, Γ α µν, Cµν) α are gven by (a) λ α µσ λ α σµ = λ ɛ R α ɛµσ + λ α ɛ Λ ɛ σµ + λ α ɛ R ɛ σµ (b) λ α µσ λ α σµ = λ ɛ S α ɛµσ + λ α ɛ T ɛ σµ (c) λ α µ σ λ α σ µ = λ ɛ P α ɛµσ + λ α ɛ C ɛ σµ + λ α ɛ P ɛ σµ, where R α νµσ : = (δ σ Γ α νµ δ µ Γ α νσ) + (Γ ɛ νµγ α ɛσ Γ ɛ νσγ α ɛµ) + L α νµσ, (h-curvature) S α νµσ : = σ C α νµ µ C α νσ + C ɛ νµc α ɛσ C ɛ νσc α ɛµ, P α νµσ : = C α νµ σ µ Γ α νσ P ɛ σµc α νɛ, gven that L α νµσ := C α νɛ R ɛ µσ and P ν σµ := µ N ν σ Γ ν µσ. (v-curvature) (hv-curvature) A drect cnsequence f the abve cmmutatn frmulae, tgether wth the fact that λ α µ = λ α µ = 0, s the fllwng Crllary 4.3. The three curvature tensrs R α νµσ, S α νµσ and P α νµσ f the canncal cnnectn D = (N α µ, Γ α µν, C α µν) vansh dentcally. It s t be nted that the abve result s a natural generalzatn f the crrespndng result f the classcal AP-gemetry [15]. The Banch denttes [4] fr the canncal d-cnnectn (N α µ, Γ α µν, C α µν) gves Prpstn 4.4. The fllwng denttes hld (a) S ν,µ,σ Λ α νµ σ = S ν,µ,σ(λ α µɛλ ɛ νσ + L α µνσ) (b) S ν,µ,σ T α νµ σ = S ν,µ,σ(t α µɛt ɛ νσ), where S ν,µ,σ dentes a cyclc permutatn n ν, µ, σ. Crllary 4.5. The fllwng denttes hld: (a) Λ ɛ µν ɛ = β µ ν β ν µ + β ɛ Λ ɛ µν + S ɛ,ν,µ L ɛ ɛνµ. (b) T ɛ µν ɛ = B µ ν B ν µ + B ɛ T ɛ µν, Prf. Bth denttes fllw by cntractng the ndces α and σ n the denttes (a) and (b) f Prpstn 4.4, takng nt accunt that β µ = Λ ɛ µɛ, B µ = T ɛ µɛ and L α µνσ = L α µσν.
128 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed In addtn t the Remannan and the canncal d-cnnectns, ur space admts at least tw ther natural d-cnnectns. In analgy t the classcal AP-space, we defne the dual d-cnnectn D = (N α µ, Γ α µν, C α µν) by (4.2) Γα µν := Γ α νµ, Cα µν := C α νµ and the symmetrc d cnnectn D = (N α µ, Γ α µν, Ĉα µν) by (4.3) Γα µν := 1 2 (Γα µν + Γ α νµ), Ĉ α µν := 1 2 (Cα µν + C α νµ). Cvarant dfferentatn wth respect t Γ α µν and Γ α µν wll be dented by and respectvely. Nw, crresndng t each f the fur d-cnnectns there are three curvature tensrs. Therefre, we have a ttal f twelve curvature tensrs three f whch, as already mentned, vansh dentcally. The vanshng f the curvature tensrs f the canncal d-cnnectn allws us t express, n a relatvely cmpact frm, sx f the ther curvature tensrs (the h- and v-curvature tensrs) crrespndng t the Remannan, symmetrc and the dual d-cnnectns. These curvature tensrs are expressed n terms f the trsn tensrs Λ α µν, Tµν α and ther cvarant dervatves wth respect t the canncal d-cnnectn, tgether wth the curvature Rµν α f the nn-lnear cnnectn Nµ α. The ther three hv-curvature tensrs are calculated, thugh ther expressns are mre cmplcated. Ths s t be expected snce the expressn btaned fr the hvcurvature tensr f the canncal d-cnnectn lacks the symmetry prpertes enjyed by the h- and v-curvature tensrs. Therem 4.6. The h-, v- and hv-curvature tensrs f the dual d-cnnectn D = (N α µ, Γ α µν, C α µν) can be expressed n the frm: (a) R α µσν = Λ α σν µ + Cα ɛµr ɛ σν + L α σνµ + L α νµσ. (b) S α µσν = T α σν µ. (c) P α νµσ = T α µν σ Λα σν µ + T ɛ µνλ α σɛ T α µɛλ ɛ σν Λ α ɛνc ɛ σµ P ɛ σµt α ɛν. The crrespndng curvature tensrs f the symmetrc d-cnnectn D = (N α µ, Γ α µν, Ĉα µν) can be expressed n the frm: (d) R α µσν = 1 2 (Λα µν σ Λα µσ ν ) + 1 4 (Λɛ µνλ α σɛ Λ ɛ µσλ α νɛ) + 1 2 (Λɛ σνλ α ɛµ) + 1 2 (T α ɛµr ɛ σν). (e) Ŝα µσν = 1 2 (T α µν σ T α µσ ν ) + 1 4 (T ɛ µνt α σɛ T ɛ µσt α νɛ) + 1 2 (T ɛ σνt α ɛµ). (f) P α νµσ = 1 2 (Λα µν σ Λα σν µ ) + 1 4 Λɛ σµt α ɛν 1 2 Λα ɛνc ɛ σµ + 1 4 S µ,ν,σλ ɛ µνλ α σɛ 1 2 P ɛ σµt α ɛν. The crrespndng curvature tensrs f the Remannan d-cnnectn D = (N α µ, Γ α µν, C α µν) can be expressed n the frm (g) R α µσν = γ α µν σ γα µσ ν + γɛ µσγ α ɛν γ ɛ µνγ α ɛσ + γ α µɛλ ɛ νσ + G α µɛr ɛ νσ.
Gemetry f parallelzable manflds 129 (h) S α µσν = G α µν σ Gα µσ ν + Gɛ µσg α ɛν G ɛ µνg α ɛσ + G α µɛtνσ. ɛ () P α νµσ = u γνσ α G α νµ σ + (Gɛ νµ Cνµ)γ ɛ ɛσ α (G α ɛµ Cɛµ)γ α νσ ɛ + PσµG ɛ α νɛ. Prf. We prve (a) and (c) nly. The prf f the ther parts s smlar. (a) We have R α µσν = δ ν Γα µσ δ σ Γα µν + Γ ɛ µσ Γ α ɛν Γ ɛ µν Γ α ɛσ + C α µɛr ɛ σν = δ ν Γ α σµ δ σ Γ α νµ + Γ ɛ σµγ α νɛ Γ ɛ νµγ α σɛ + C α ɛµr ɛ σν = {δ ν Γ α σµ + Γ ɛ σµ(λ α νɛ + Γ α ɛν)} {δ σ Γ α νµ + Γ ɛ νµ(λ α σɛ + Γ α ɛσ)} + C α ɛµr ɛ σν = (δ ν Γ α σµ + Γ ɛ σµγ α ɛν) (δ σ Γ α νµ + Γ ɛ νµγ α ɛσ) (Γ ɛ σµλ α ɛν + Γ ɛ νµλ α σɛ) + C α ɛµr ɛ σν = (R α σµν C α σɛr ɛ µν + δ µ Γ α σν + Γ ɛ σνγ α ɛµ) (R α νµσ C α νɛr ɛ µσ + δ µ Γ α νσ + Γ ɛ νσγ α ɛµ) (Γ ɛ σµλ α ɛν + Γ ɛ νµλ α σɛ) + C α ɛµr ɛ σν. = δ µ Λ α σν + Γ α ɛµλ ɛ σν Γ ɛ σµλ α ɛν Γ ɛ νµλ α σɛ + C α ɛµr ɛ σν + C α σɛr ɛ νµ + C α νɛr ɛ µσ = Λ α σν µ + Cα ɛµr ɛ σν + L α σνµ + L α νµσ. (c) We have P α νµσ = C α µν σ µ Γ α σν ( µ N ɛ σ Γ ɛ σµ)c α ɛν = C α νµ σ + (Cα µν σ Cα νµ σ ) µ Λ α σν µ Γ α νσ µ N ɛ σ(t α ɛν + C α νɛ) + (Λ ɛ σµ + Γ ɛ µσ)(t α ɛν + C α νɛ) = P α νµσ ( µ N ɛ σ Γ ɛ µσ)t α ɛν µ Λ α σν + Λ ɛ σµc α ɛν + (C α µν σ Cα νµ σ ) = (C α µν σ Cα νµ σ ) + Λɛ σµc α ɛν µ Λ α σν P ɛ σµt α ɛν = T α µν σ + Cɛ µνλ α σɛ C α µɛλ ɛ σν µ Λ α σν P ɛ σµt α ɛν = T α µν σ µ Λ α σν + (T ɛ µν + C ɛ νµ)λ α σɛ (T α µɛ + C α ɛµ)λ ɛ σν P ɛ σµt α ɛν = T α µν σ Λα σν µ + T ɛ µνλ α σɛ T α µɛλ ɛ σν Λ α ɛνc ɛ σµ P ɛ σµt α ɛν. whch are the requred frmulae. 5 Fundamental secnd rank tensrs Due t the mprtance f secnd rder symmetrc and skew-symmetrc tensrs n physcal applcatns, we here lst such tensrs n Table 2 belw. We regard these tensrs as fundamental snce ther cunterparts n the classcal AP-cntext play a key rle n physcal applcatns. Mrever, n the AP-gemetry, mst secnd rank tensrs whch have physcal sgnfcance can be expressed as a lnear cmbnatn f these fundamental tensrs. The Table s cnstructed as smlar as pssble t that gven by Mkhal (cf. [5], Table 2), t facltate cmparsn wth the case f the classcal AP-gemetry whch has many physcal applcatns [14]. Crrespndng hrzntal
130 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed and vertcal tensrs are dented by the same symbl wth the vertcal tensrs barred. It s t be nted that all vertcal tensrs have n cunterpart n the classcal AP-cntext. Table 2. Summary f the fundamental symmetrc and skew-symmetrc secnd rank tensrs Hrzntal Vertcal Skew-Symmetrc Symmetrc Skew-Symmetrc Symmetrc ξµν := γµν α α ξ µν := Gµν α α γµν := βαγµν α γ µν := BαGµν α ηµν := βɛ Λ ɛ µν φµν := βɛ Ω ɛ µν η µν := Bɛ T ɛ µν φ µν := Bɛ D ɛ µν χµν := Λ α µν α ψµν := Ω ɛ µν ɛ χ µν := T α µν α ψ µν := D α µν α ɛµν := 1 2 (β µ ν β ν µ ) θµν := 1 2 (β µ ν + β ν µ ) ɛµν := 1 2 (B µ ν B ν µ ) θµν := 1 2 (B µ ν + B ν µ ) k µν := h µν := k µν := h µν := γ ɛ αµγ α νɛ γ ɛ µαγ α ɛν γ ɛ αµγ α νɛ + γ ɛ µαγ α ɛν G ɛ αµg α νɛ G ɛ µαg α ɛν G ɛ αµg α νɛ + G ɛ µαg α ɛν σµν := γ ɛ αµ γα ɛν σµν := G ɛ αµ Gα ɛν ωµν := γ ɛ µα γα νɛ ωµν := G ɛ µα Gα νɛ αµν := βµβν αµν := BµBν Due t the metrcty cndtn n Therem 3.3, ne can use the metrc tensr g µν and ts nverse g µν t perfrm the peratns f lwerng and rasng tensr ndces under the h- and v- cvarant dervatves relatve t the canncal d-cnnectn. Thus, cntractn wth the metrc tensr f the abve fundamental tensrs gves the fllwng table f scalars: Table 3. Summary f the fundamental scalars Hrzntal α := β µβ µ θ := β µ µ φ := β ɛ Ω ɛµ µ ψ := Ω αµ µ α ω := γ ɛµ α γ α µɛ σ := γ ɛ α µ γ α ɛµ h := 2γ αµ ɛ γ ɛ αµ Vertcal α := B µ B µ θ := B µ µ φ := B ɛ D ɛµ µ ψ := D αµ µ α ω := G ɛµ α G α µɛ σ := G ɛ α µ G α ɛµ h := 2G αµ ɛ G ɛ αµ In physcal applcatns, secnd rder symmetrc tensrs f zer trace have specal mprtance. Fr example, n the case f electrmagnetsm, the tensr characterzng the electr-magnetc energy s a secnd rder symmetrc tensr havng zer trace. S t s f nterest t search fr such tensrs. The Table belw gves sme f the secnd rank tensrs f zer trace.
Gemetry f parallelzable manflds 131 Table 4. Summary f the fundamental tensrs f zer trace Hrzntal Vertcal φ µν + 2α µν φ µν + 2ᾱ µν ψ µν + 2θ µν ψ µν + 2θ µν h µν + 2ω µν h µν + 2ω µν 1 2 (φµν ψµν) + θµν αµν 1 2 gµνβα 1 α 2 (φ µν ψ µν ) + θµν αµν 1 2 gµνbα α We nw cnsder sme useful secnd rank tensrs whch are nt expressble n terms f the fundamental tensrs appearng n Table 2. Unlke the tensrs f Table 2, sme f the tensrs t be defned belw have n hrzntal and vertcal cunterparts. T ths end, let L µν := L α αµν = C α αɛr ɛ µν, M µν := L α µαν = C α µɛ R ɛ αν, N µν := C α ɛµ R ɛ αν, F µν := C α ɛµ R ɛ αν. Then, clearly T µν := M µν N µν = T α µɛ R ɛ αν, G µν := M µν F µν = G α µɛ R ɛ αν, G µν T µν = G α ɛµ R ɛ αν. Fnally, let T := g µν T µν and G := g µν G µν. By the abve, we have the fllwng: Symmetrc secnd rank tensrs: M (µν), N (µν), F (µν). Skew-symmetrc secnd rank tensrs: M [µν], N [µν], F [µν], L µν. 6 Cntracted curvatures and curvature scalars It may be cnvenent, fr physcal reasns, t cnsder secnd rank tensrs derved frm the curvature tensrs by cntractns. It s als f nterest t reduce the number f these tensrs t a mnmum whch s fundamental (cf. Prpstns 6.1 and 6.2). Cntractng the ndces α and µ n the expressns btaned fr the h- and v- curvature tensrs n Therem 4.6, takng nt accunt Crllary 4.5, we btan Prpstn 6.1. Let R σν := R ασν, α Rσν := R ασν α and R σν := R α ασν wth smlar expressns fr S σν, Ŝσν and S σν. Then, we have (a) R σν = β σ ν β ν σ + β ɛ Λ ɛ σν + B ɛ R ɛ σν, (b) S σν = B σ ν B ν σ + B ɛ T ɛ σν, (c) R σν = 1 2 R σν, (d) Ŝσν = 1 2 S σν,
132 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed (e) R σν = S σν = 0. Prpstn 6.2. Let R µσ := R µσα, α R µσ := R µσα α and R µσ := R α µσα wth smlar expressns fr S µσ, Ŝµσ and S µσ. Then, we have (a) R µσ = β σ µ + CɛµR α σα ɛ + L α σαµ + L α αµσ, (b) S µσ = B σ µ, (c) R µσ = 1 R 2 µσ + 1 4 {β ɛλ ɛ σµ + Λ ɛ ασλ α µɛ}, (d) Ŝµσ = 1 S 2 µσ + 1 4 {B ɛtσµ ɛ + TασT ɛ µɛ}, α (e) R µσ = β µ σ γµσ α α + β ɛγµσ ɛ γµɛγ α σα ɛ + G α µɛrασ, ɛ (f) S µσ := S α µσα = B µ σ G α µσ α + B ɛg ɛ µσ G α µɛg ɛ σα. Prpstn 6.3. The fllwng hlds. (a) R [µσ] = 1 2 {β σ µ β µ σ } + C ɛ ɛαr α µσ + C ɛ (ασ) Rα ɛµ C ɛ (αµ) Rα ɛσ, (b) R (µσ) = 1 2 {β σ µ + β µ σ + T ɛ αµr α σɛ + T ɛ ασr α µɛ}, (c) S [µσ] = 1 2 {B σ µ B µ σ }, (d) S (µσ) = 1 2 {B σ µ + B µ σ }, (e) R [µσ] = 1 2 R [µσ] + 1 4 β ɛ Λ ɛ σµ, (f) R (µσ) = 1 2 R (µσ) + 1 4 Λɛ ασ Λ α µɛ, (g) Ŝ[µσ] = 1 2 S [µσ] + 1 4 B ɛ T ɛ σµ, (h) Ŝ(µσ) = 1 2 S (µσ) + 1 4 T ɛ ασ T α µɛ, () R [µσ] = 1 2 {Lα αµσ + C α σɛ R ɛ αµ C α µɛ R ɛ ασ}, (j) R (µσ) = 1 2 {(β µ σ + β σ µ ) Ω α µσ α + β ɛ Ω ɛ µσ} γµɛ α γσα ɛ + 1 2 {Gα µɛ Rασ ɛ + G α σɛ Rαµ}, ɛ (k) S [µσ] = 0, (l) S (µσ) = 1 2 {(B µ σ + B σ µ ) Dµσ α α + B ɛ Dµσ} ɛ G α µɛ G ɛ σα. Crllary 6.4. The fllwng hlds: (a) R σ σ := g µσ Rµσ = β σ σ + T ɛσ α R α ɛσ, (b) S σ σ := g µσ Sµσ = B σ σ,
Gemetry f parallelzable manflds 133 (c) R σ σ := g µσ Rµσ = 1 2 {βσ σ + T ɛσ α R α ɛσ} + 1 4 Λɛσ α Λ α ɛσ, (d) Ŝσ σ := g µσ Ŝ µσ = 1 2 Bσ σ + 1 4 T ɛσ α T α ɛσ, (e) R σ σ := g µσ R µσ = β σ σ 1 2 Ωασ σ α + 1 2 β α Ω ασ σ γ ασ ɛ γ ɛ σα + G ασ ɛ R ɛ ασ, (f) S σ σ := g µσ S µσ = B σ σ 1 2 Dασ σ α + 1 2 B α D ασ σ G ασ ɛ G ɛ σα. We nw apply a dfferent methd fr calculatng bth R µσ and S µσ, nw expressed n terms f the cvarant dervatve f the cntrsn tensrs wth respect t the Remannan d-cnnectn. Then we btan Prpstn 6.5. The Rcc tensrs R µσ and S µσ can be expressed n the frm (a) R µσ = β µ σ γ α µσ α β ɛγ ɛ µσ + γ ɛ µαγ α ɛσ + G α µɛr ɛ ασ. (b) S µσ = B µ σ G α µσ α B ɛg ɛ µσ + G ɛ µαg α ɛσ. Prf. We prve (a) nly; the prf f (b) s smlar. We have 0 = R α µσα = (δ α Γ α µσ δ σ Γ α µα) + (Γ ɛ µσγ α ɛα Γ ɛ µαγ α ɛσ) + R ɛ σαc α µɛ = δ α ( Γ α µσ + γ α µσ) δ σ ( Γ α µα + γ α µα) + ( Γ ɛ µσ + γ ɛ µσ)( Γ α ɛα + γ α ɛα) ( Γ ɛ µα + γ ɛ µα)( Γ α ɛσ + γ α ɛσ) + R ɛ σαc α µɛ = R µσ (δ σ γµα α γɛα α Γ ɛ µσ) + (δ α γµσ α + γµσ ɛ Γ α ɛα γɛσ α Γ ɛ µα γµɛ α Γ ɛ σα) + Rσα(C ɛ µɛ α C α µɛ) + γµσγ ɛ ɛα α γµαγ ɛ ɛσ. α Cnsequently, whch s the requred frmula. R µσ = β µ σ γ α µσ α β ɛγ ɛ µσ + γ ɛ µαγ α ɛσ + G α µɛr ɛ ασ, In vew f Prpstn 6.2 (e) and (f) and Prpstn 6.5, we btan Crllary 6.6. The fllwng denttes hlds: (a) (β µ σ β µ σ ) (γ α µσ α γα µσ α) = (γɛ µαω α σɛ 2β ɛ γ ɛ µσ) (b) (B µ σ B µ σ ) (G α µσ α Gα µσ α) = (Gɛ µαd α σɛ 2B ɛ G ɛ µσ). The next tw tables summarze the results btaned n ths sectn, where the cntracted curvatures are expressed n terms f the fundamental tensrs.
134 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed Table 5 (a). Secnd rank curvature tensrs Skew-symmetrc Symmetrc Dual R[µσ] = ɛ σµ L σµ + M [σµ] + N [σµ] R(µσ) = θ µσ + M (µσ) N (µσ) S [µσ] = ɛ σµ S(µσ) = θ µσ Symmetrc R[µσ] = 1 2 R [µσ] + 1 4 η σµ R(µσ) = 1 2 R (µσ) + 1 4 {h µσ ω µσ σ µσ } Remannan Ŝ [µσ] = 1 2 S [µσ] + 1 4 η σµ R [µσ] = 1 2 L µσ F [µσ] Ŝ (µσ) = 1 2 S (µσ) + 1 4 {h µσ ω µσ σ µσ } R (µσ) = θ µσ 1 2 (ψ µσ φ µσ ) S [µσ] = 0 ω µσ + M (µσ) F (µσ) S (µσ) = θ µσ 1 2 (ψ µσ φ µσ ) ω µσ Table 5 (b). h- and v-scalar curvature tensrs h-scalar curvature v-scalar curvature Dual Rσ σ = θ + T Sσ σ = θ Symmetrc Rσ σ = 1 2 (θ + T ) 1 4 (3ω + σ) Ŝσ σ = 1 2 θ 1 4 (3ω + σ) Remannan R σ σ = θ 1 2 (ψ φ) ω + G S σ σ = θ 1 2 (ψ φ) ω 7 The W -tensrs The W -tensr was frst defned by M. Wanas n 1975 [13] and has been used by F. Mkhal and M. Wanas [6] t cnstruct a gemetrc thery unfyng gravty and electrmagnetsm. Recently, tw f the authrs f ths paper studed sme f the prpertes f ths tensr n the cntext f the classcal AP-space [15]. Defntn 7.1. Let (M, λ) be a generalzed AP-space. Fr a gven d-cnnectn D = (N α β, Γα µν, C α µν), the hrzntal W -tensr (hw -tensr) H α µνσ s defned by the frmula λ µ νσ λ µ σν = λ ɛ H ɛ µνσ, whereas the vertcal W -tensr (vw -tensr) V α µνσ s defned by the frmula λ µ νσ λ µ σν = λ ɛ V ɛ µνσ, where and are the hrzntal and the vertcal cvarant dervatves wth respect t the cnnectn D.
Gemetry f parallelzable manflds 135 We nw carry ut the task f calculatng the dfferent W -tensrs. As ppsed t the classcal AP-space, whch admts essentally ne W -tensr crrespndng t the dual cnnectn, we here have 4 dstnct W -tensrs: the hrzntal and vertcal W -tensrs crrespndng t the dual d-cnnectn, the hrzntal W -tensr crrespndng t the symmetrc d-cnnectn and, fnally, the hrzntal W -tensr crrespndng t the Remannan d-cnnectn. The remanng W -tensrs cncde wth the crrespndng curvature tensrs. It s t be nted that sme f the expressns btaned fr the W -tensrs are relatvely mre cmpact than thse btaned fr the crrespndng curvature tensrs. Therem 7.2. The hw-tensr H µνσ, α the vw-tensr Ṽ µνσ, α the hw-tensr Ĥα µνσ and the hw-tensr H α µνσ crrespndng t the dual, symmetrc and the Remannan d- cnnectns respectvely can be expressed n the frm: (a) H α µνσ = Λ α σν µ + Λɛ νσλ α µɛ + S µ,ν,σ L α µσν. (b) Ṽ α µνσ = T α σν µ + T ɛ νσt α µɛ. (c) Ĥα µνσ = 1 2 (Λα µν σ Λα µσ ν ) + 1 4 (Λɛ µνλ α σɛ Λ ɛ µσλ α νɛ) + 1 2 (Λɛ σνλ α ɛµ). (d) H α µνσ = γ α µν σ γα µσ ν + γɛ µσγ α ɛν γ ɛ µνγ α ɛσ + Λ ɛ νσγ α µɛ. Prf. We prve (a) nly. The prf f the ther parts s smlar. We have λ ɛ Hɛ µνσ = λ ɛ Rɛ µσν + λ µ ɛ Λɛ σν + λ µ ɛ R ɛ σν. Hence, takng nt accunt Therem 4.6 (a), we btan H µνσ α = R µσν α + λ α (δ ɛ λ µ λ β Γ β ɛµ) Λ ɛ σν + λ α ( ɛ λ µ λ β C ɛµ)r β σν ɛ = R α µσν + Λ ɛ νσ(γ α µɛ Γ α ɛµ) + R ɛ σν(c α µɛ C α ɛµ) = Λ α σν µ + Cα ɛµr ɛ σν + L α σνµ + L α νµσ + Λ ɛ νσλ α µɛ + T α µɛr ɛ σν = Λ α σν µ + T α ɛµr ɛ σν + C α µɛr ɛ σν + L α σνµ + L α νµσ + Λ ɛ νσλ α µɛ + T α µɛr ɛ σν = Λ α σν µ + Λɛ νσλ α µɛ + S µ,ν,σ L α µσν. Prpstn 7.3. Let H νσ := H ανσ, α Ĥ νσ := Ĥα ανσ and expressn fr Ṽνσ. Then, we have H νσ := H α ανσ wth smlar (a) H νσ = β σ ν β ν σ + 2β ɛ Λ ɛ σν, (b) Ṽνσ = B σ ν B ν σ + 2B ɛ T ɛ σν, (c) Ĥνσ = 1 2 { H νσ + β ɛ Λ ɛ νσ}, (d) H νσ = 0.
136 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed Prpstn 7.4. Let H µσ := H α µασ, Ĥµσ := Ĥα µασ and expressns fr Ṽµσ. Then, we have H µσ := H α µασ wth smlar (a) H µσ = β σ µ + Λ ɛ ασλ α µɛ + S α,µ,σ L α αµσ, (b) Ṽµσ = B σ µ + TασT ɛ µɛ, α (c) Ĥµσ = 1 H 2 µσ + 1 4 (β ɛλ ɛ σµ + Λ ɛ σαλ α µɛ), (d) H µσ = β µ σ γµσ α α + β ɛγµσ ɛ γσαγ ɛ µɛ. α Prpstn 7.5. The fllwng hlds: (a) H [µσ] = 1 2 {β σ µ β µ σ } + S α,µ,σ L α αµσ, (b) H (µσ) = 1 2 {β σ µ + β µ σ } + Λ ɛ ασλ α µɛ, (c) Ṽ[µσ] = 1 2 {B σ µ B µ σ }, (d) Ṽ(µσ) = 1 2 {B σ µ + B µ σ } + TασT ɛ µɛ, α (e) Ĥ[µσ] = 1 H 2 [µσ] + 1 4 β ɛλ ɛ σµ, (f) Ĥ(µσ) = 1 H 2 (µσ) + 1 4 Λɛ σαλ α µɛ, (g) H [µσ] = 1 2 S αµσl α αµσ, (h) H (µσ) = 1 2 {(β µ σ + β σ µ ) Ω α µσ α + β ɛω ɛ µσ} γµɛγ α σα. ɛ Crllary 7.6. the fllwng hlds: (a) H α α = β α α + Λ ɛµ αλ α ɛµ, (b) Ṽ α α = B α α + T ɛµ αt α ɛµ, (c) Ĥα α = 1 2 βα α + 1 4 Λɛµ αλ α ɛµ, (d) H σ σ = β σ σ 1 2 Ωασ σ α + 1 2 β αω ασ σ γ ασ ɛγ ɛ σα. Takng nt accunt Prpstn 4.4, Therem 7.2 and the Banch dentty [4] fr the Remannan d-cnnectn, we get the fllwng Prpstn 7.7. The hw-tensrs H µνσ, α Ĥµνσ, α satsfy the fllwng denttes: H α µνσ and the vw-tensrs Ṽ α µνσ (a) S µ,ν,σ Hα µνσ = 2S µ,ν,σ (Λ α µɛ Λ ɛ νσ + L α µσν). (b) S µ,ν,σ Ṽ α µνσ = 2S µ,ν,σ (T α µɛ T ɛ νσ). (c) S µ,ν,σ Ĥ α µνσ = S µ,ν,σ L α µσν.
Gemetry f parallelzable manflds 137 (d) S µ,ν,σ H α µνσ = S µ,ν,σ L α µσν. We cllect the results btaned n ths sectn n the fllwng tables, where the cntracted W -tensrs are expressed n terms f the fundamental tensrs. Table 6 (a). Secnd rank W-tensrs Skew-symmetrc Symmetrc Dual H[µσ] = ɛ σµ L σµ + 2M [σµ] H(µσ) = θ µσ (ω µσ + σ µσ h µσ) Ṽ [µσ] = ɛ σµ Ṽ (µσ) = θ µσ ( ω µσ + σ µσ h µσ ) Symmetrc Ĥ [µσ] = 1 H 2 [µσ] + 1 4 ησµ Ĥ (µσ) = 1 H 2 (µσ) + 1 {ωµσ + σµσ hµσ} 4 Remannan H [µσ] = 1 2 L µσ M [µσ] H (µσ) = θ µσ 1 2 (ψ µσ φ µσ ) ω µσ Table 6 (b). Scalar W-tensrs h-scalar W-tensrs v-scalar W-tensrs Dual Hσ σ = θ (3ω + σ) Ṽ σ σ = θ (3 ω + σ) Symmetrc Ĥσ σ = 1 2 θ 1 4 (3ω + σ) Remannan H σ σ = θ 1 2 (ψ φ) ω Cncludng remarks In the present artcle, we have develped a parallelzable structure n the cntext f a generalzed Lagrange space. Fur dstngushed cnnectns, dependng n ne nnlnear cnnectn, are used t explre the prpertes f ths space. Dfferent curvature tensrs characterzng ths structure are calculated. The cntracted curvature tensrs necessary fr physcal applcatns are gven and cmpared (Tables 5(a)). The traces f these tensrs are derved and cmpared (Table 5(b)). Fnally, the dfferent W - tensrs wth ther cntractns and traces are als derved (Tables 6(a) and 6(b)).
138 M.I. Wanas, Nabl L. Yussef and A.M. Sd-Ahmed On the present wrk, we have the fllwng cmments and remarks: 1. Exstng theres f gravty suffer frm sme prblems cnnected t recent bserved astrphyscal phenmena, especally thse admttng anstrpc behavr f the system cncerned (e.g. the flatness f the rtatn curves f spral galaxes). S, theres n whch the gravtatnal ptental depends n bth pstn and drectn are needed. Such theres are t be cnstructed n spaces admttng ths dependence. Ths s ne f the ams mtvatng the present wrk. 2. Amng the advantages f the AP-gemetry are the ease n calculatns and the dverse and ts thrugh applcatns. In ths wrk, we have kept as clse as pssble t the classcal AP-case. Hwever, the extra degrees f freedm n ur GAP-gemetry have created an abundance f gemetrc bjects whch have n cunterpart n the classcal AP-gemetry. Snce the physcal meanng f mst f the gemetrc bjects f the classcal AP-structure s clear, we hpe t attrbute physcal meanng t the new gemetrc bjects appearng n the present wrk, especally the vertcal quanttes. 3. Due t the wealth f the GAP-gemetry, ne s faced wth the prblem f chsng gemetrc bjects that represent true physcal quanttes. As a frst step t slve ths prblem, we have wrtten all secnd rder tensrs n terms f the fundamental tensrs defned n sectn 5. Ths s dne t facltate cmparsn between these tensrs and t be able t chse the mst apprprate fr physcal applcatn. The same prcedure has been used fr scalars. 4. The paper s nt ntended t be an end n tself. In t, we try t cnstruct a gemetrc framewrk capable f dealng wth and descrbng physcal phenmena. The success f the classcal AP-gemetry n physcal applcatns made us chse ths gemetry as a gude lne. The physcal nterpretatn f the gemetrc bjects exstng n the GAPgemetry and nt n the AP-gemetry wll be further nvestgated n a frthcmng paper. Ths paper s nt an end n tself, but rather the begnnng f a research drectn. The physcal nterpretatn f the gemetrc bjects n the GAP-space that have n cunterpart n the classcal AP-space wll be further nvestgated n frthcmng papers. Acknwledgement. Ths paper was presented n The Internatnal Cnference n Fnsler Extensns f Relatvty Thery held at Car, Egypt, Nvember 4-10, 2006. ArXv Number: 0704.2001. References [1] D. Ba, S. Chern and Z. Shen, An ntrductn t Remann-Fnsler Gemetry, Graduate Texts n Mathematcs, Sprnger, 2000. [2] F. Brckell and R. S Clark, Dfferentable manflds, Van Nstrand Renhld C., 1970.
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