Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces

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1 1 Gemetry f Parallelzable Manflds n the Cntext f Generalzed Lagrange Spaces arxv: v1 [gr-qc] 16 Apr 2007 M. I. Wanas, N. L. Yussef and A. M. Sd-Ahmed Department f Astrnmy, Faculty f Scence, Car Unversty wanas@frcu.eun.eg Department f Mathematcs, Faculty f Scence, Car Unversty nyussef@frcu.eun.eg, amrs@maler.eun.eg 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 (TM) (the pullback f the tangent bundle TM by π : TM 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 (TM) 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. Ths paper, hwever, 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. 1 Keywrds: Parallelzable manfld, Generalzed Lagrange space, AP-gemetry, GAPgemetry, Canncal d-cnnectn, W-tensr AMS Subject Classfcatn. 53B40, 53A40, 53B50. 1 Ths paper was presented n The Internatnal Cnference n Fnsler Extensns f Relatvty Thery held at Car, Egypt, Nvember 4-10, 2006.

2 2 1. Intrductn The gemetry f parallelzable manflds r the abslute parallelsm gemetry (AP-gemetry) ([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 machnary 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 (TM) (the pullback f the tangent bundle TM by the prjectn π : TM 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 (TM). 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

3 3 expressed n terms f the trsn tensrs f the canncal d-cnnectn. Cntractns 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 : TM 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 TM. The crdnate transfrmatn law n TM s gven by: x µ = x µ (x ν ), y µ = p µ ν yν, where p µ ν = xµ and det(p µ x ν ν ) 0. Defntn 2.1. A nn-lnear cnnectn N n TM 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 where p ǫ β σ = pǫ β x σ = 2 x ǫ x β x σ. N α β = pα α p β β N α β + p α ǫ p ǫ β σ yσ, (2.1) The nn-lnear cnnectn N leads t the drect sum decmpstn T u (TM) = H u (TM) V u (TM), u T M = TM \ {0}, where H u (TM) s the hrzntal space at u asscated wth N supplementary t the vertcal space V u (TM). If δ µ := µ Nµ α α, where µ :=, x µ µ :=, then ( y µ µ ) s the natural bass f V u (TM) and (δ µ ) s the natural bass f H u (TM) adapted t N. Defntn 2.2. A dstngushed cnnectn (d-cnnectn) n M s a trplet D = (Nµ α, Γ α µν, Cµν), α where Nµ α (x, y) s a nn-lnear cnnectn n TM and Γ α µν(x, y) and Cµν α (x, y) transfrm accrdng t the fllwng laws: Γ α µ ν = pα α pµ µ p ν ν Γα µν + pα ǫ pǫ µ ν, (2.2) C α µ ν = pα α pµ µ p ν ν Cα µν. (2.3) In ther wrds, Γ α µν transfrm as the ceffcents f a lnear cnnectn, whereas Cα 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: A α µ ν := δ νa α µ + Aǫ µ Γα ǫν Aα ǫ Γǫ µν ; (2.4) A α µ ν := ν A α µ + Aǫ µ Cα ǫν Aα ǫ Cǫ µν. (2.5) 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.

4 4 Defntn 2.5. Let (M, g) be a generalzed Lagrange space equpped wth a nn-lnear 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 g µν α = 0, g µν α = 0. (2.6) 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 TM. 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: Γ α µν = 1 2 gαǫ (δ µ g νǫ + δ ν g µǫ δ ǫ g µν ), (2.7) C α µν = 1 2 gαǫ ( µ g νǫ + ν g µǫ ǫ g µν ). (2.8) 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 (TM) admts n glbal lnearly ndependent sectns (π-vectr felds) λ(x, y), = 1,..., n. If λ = ( λ α ), α = 1,..., n, then λ α λ β = δβ α, where ( λ α ) dentes the nverse f the matrx ( λ α ). λ α λ α = δ j, (3.1) j 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. 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.

5 5 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 g µν (x, y) := λ µ λ ν, (3.2) defnes a metrc n the pull-back bundle π 1 (TM) wth nverse gven by g µν (x, y) = λ µ λ ν (3.3) Assume that M s a GAP-space equpped wth a nn-lnear cnnectn Nµ α. By Therem 2.6, there exsts n (M, g) a unque trsn-free metrcal d-cnnectn D = (Nµ α, 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: Γ α µν := Γ α µν + λ α λ µ ν, (3.4) Cµν α := C α µν + λ α λ µ ν. (3.5) 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 λ α µ = 0, λ α µ = 0. (3.6) Ths can be shwn as fllws: λ α µ = δ µ λ α + λ ǫ Γ α ǫµ = δ µλ α + λ ǫ ( Γ α ǫµ + λ α λ j j ǫ µ ) = (δ µ λ α + λ ǫ Γ α ǫµ) λ α j µ ( λ ǫ λ ǫ ) = 0. In exactly the same way, t can be shwn that j λ α µ = 0. Hence, we btan the fllwng 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 = (Nα µ, Γα µν, Cα µν ) s explctly expressed n terms f λ n the frm Γ α µν = λ α (δ ν λ µ ), Cµν α = λ α ( ν λ µ ). (3.7) Prf. Snce λ α ν = 0, we have δ ν λ α = λ ǫ Γ α ǫν. Multplyng bth sdes by λ µ, takng nt accunt the fact that λ α λ µ = δµ α, we get Γα µν = λ µ (δ ν λ α ) = λ α (δ ν λ µ ). The 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

6 6 Crllary 3.5. The Remannan d-cnnectn D = (Nµ α, Γ α µν, C α µν ) s explctely expressed n terms f λ n the frm Γ α µν = λ α (δ ν λ µ λ µ ν ), C α µν = λ α ( ν λ µ λ µ ν ). (3.8) Remark 3.6. As 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 λ α ( ν λ µ ) = p α α pµ µ p ν ν λ α ( ν λ µ ) + p α ǫ pǫ µ ν + pα α pµ µ p ν ν ǫ C α yǫ µν. (3.9) 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 γµν α := λ α λ µ ν = Γ α µν Γ α µν, Gα µν := λ α λ µ ν = Cµν α C α µν. (3.10) In analgy t the AP-space, we refer t γ α µν and G α µν as the h- and v-cntrtn tensrs respectvely. Let Λ α µν := Γα µν Γα νµ = γα µν γα νµ. (3.11) be the trsn tensr f the canncal cnnectn Γ α µν and Ω α µν := γµν α + γνµ. α (3.12) Smlarly, let Tµν α := Cµν α Cνµ α = G α µν G α νµ (3.13) be what we may call the trsn tensr f Cµν α and D α µν := Gα µν + Gα νµ. (3.14) Nw, f γ σµν := g ǫσ γ ǫ µν and G σµν := g ǫσ G ǫ µν, then γ σµν and G σµν are skew symmetrc n the frst par f ndces. Ths, n turn, mples that Hence, f then γǫν ǫ = Gǫ ǫν = 0. (3.15) β µ := γ ǫ µǫ, B µ := G ǫ µǫ, Λ ǫ µǫ = γ ǫ µǫ = β µ, T ǫ µǫ = G ǫ µǫ = B µ. (3.16) 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 γ µνσ = 1 2 (Λ µνσ + Λ σνµ + Λ νσµ ) (3.17) G µνσ = 1 2 (T µνσ + T σνµ + T νσµ ), (3.18)

7 7 where Λ µνσ := g ǫµ Λ ǫ νσ and T µνσ := g ǫµ Tνσ ǫ. 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. The next table 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. 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 g µν (x) = λ µ (x) λ ν (x) g µν (x, y) = λ µ (x, y) λ ν (x, y) Remannan cnnectn Γ α µν = 1 2 gαǫ { µ g νǫ + ν g µǫ + ǫ g µν } Γ α µν = 1 2 gαǫ {δ µ g νǫ + δ ν g µǫ + δ ǫ g µν } (h) C α µν = 1 2 gαǫ { µ g νǫ + ν g µǫ + ǫ g µν } (v) Canncal cnnectn Γ α µν = λ α ( ν λ µ ) Γ α µν = λ α (δ ν λ µ ) (h-cunterpart) Cµν α = λ α ( ν λ µ ) (v-cunterpart) AP-cndtn λ α µ = 0 λ α 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 = C α µν C α νµ (v-cunterpart) Cntrsn γ α µν = Γα µν Γ α µν γ α µν = Γα µν Γ α νµ (h-cunterpart) G α µν = Cα µν C α µν (v-cunterpart) Basc vectr β µ = Λ α µα = γ α µα β µ = Λ α µα = γ α µα (h-cunterpart) B µ = T α µα = Gα µα (v-cunterpart)

8 8 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. 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 [δ σ, δ µ ] = R ǫ σµ ǫ, [δ σ, µ ] = ( µ N ǫ σ ) ǫ, (4.1) where R α σµ := δ µn α σ δ σn α µ s the curvature tensr f the nn-lnear cnnectn Nα µ. Therem 4.2. The three cmmutatn frmulae f cnnectn D = (Nµ α, Γ α µν, Cµν) α are gven by (a) λ α µσ λ α σµ = λ ǫ R α ǫµσ + λα ǫ Λ ǫ σµ + λα ǫ R ǫ σµ (b) λ α µσ λ α σµ = λ ǫ S α ǫµσ + λα ǫ T ǫ σµ (c) λ α µ σ λ α σ µ = λ ǫ P α ǫµσ + λ α ǫ C ǫ σµ + λ α ǫ P ǫ σµ, λ α crrespndng t the canncal where Rνµσ α : = (δ σγ α νµ δ µγ α νσ ) + (Γǫ νµ Γα ǫσ Γǫ νσ Γα ǫµ ) + Lα νµσ, (h-curvature) Sνµσ α : = σ Cνµ α µ Cνσ α + Cǫ νµ Cα ǫσ Cǫ νσ Cα ǫµ, (v-curvature) Pνµσ α : = Cνµ σ α µ Γ α νσ PσµC ǫ νǫ, α (hv-curvature) gven that L α νµσ := C α νǫ R ǫ µσ and P ν σµ := µ N ν σ Γ ν µσ. 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 Rνµσ α, Sα νµσ and P νµσ α cnnectn D = (Nµ α, Γα µν, Cα µν ) vansh dentcally. f the canncal 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, Γα µν, Cα µν ) gves Prpstn 4.4. The fllwng denttes hld (a) S ν,µ,σ Λ α νµ σ = S ν,µ,σ(λ α µǫ Λǫ νσ + Lα µνσ ) (b) S ν,µ,σ T α νµ σ = S ν,µ,σ(t α µǫ T ǫ νσ ), where S ν,µ,σ dentes a cyclc permutatn n ν, µ, σ. Crllary 4.5. The fllwng denttes hld: (a) Λ ǫ µν ǫ = β µ ν β ν µ + β ǫ Λ ǫ µν + S ǫ,ν,µl ǫ ǫνµ. (b) T ǫ µν ǫ = B µ ν B ν µ + B ǫ T ǫ µν,

9 9 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 ǫ µǫ, B µ = T ǫ µǫ and L α µνσ = Lα µσν. In addtn t the Remannan and the cannncal 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 and the symmetrc d cnnectn D = (N α µ, Γ α µν, Ĉα µν ) by Γ α µν := Γα νµ, Cα µν := C α νµ (4.2) Γ α µν := 1 2 (Γα µν + Γα νµ ), Ĉα µν := 1 2 (Cα µν + Cα νµ ). (4.3) 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 nnlnear 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 hv-curvature 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 = (Nµ α, Γ α µν, C µν α ) can be expressed n the frm: (a) R α µσν = Λα σν µ + Cα ǫµ Rǫ σν + Lα σνµ + Lα νµσ. (b) S α µσν = T α σν µ. (c) P α νµσ = T α µν σ Λα σν µ + T ǫ µνλ α σǫ T α µǫλ ǫ σν Λ α ǫνc ǫ σµ P ǫ σµt α ǫν. 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 ) (Λǫ µν Λα σǫ Λǫ µσ Λα νǫ ) (Λǫ σν Λα ǫµ ) (T α ǫµ Rǫ σν ). (e) Ŝα µσν = 1 2 (T α µν σ T α µσ ν ) (T ǫ µνt α σǫ T ǫ µσt α νǫ) (T ǫ σνt α ǫµ). (f) P α νµσ = 1 2 (Λα µν σ Λα σν µ ) Λǫ σµt α ǫν 1 2 Λα ǫνc ǫ σµ S µ,ν,σλ ǫ µνλ α σǫ 1 2 P ǫ σµt α ǫν.

10 10 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 = γα µν σ γα µσ ν + γǫ µσ γα ǫν γǫ µν γα ǫσ + γα µǫ Λǫ νσ + Gα µǫ Rǫ νσ. (h) S α µσν = Gα µν σ Gα µσ ν + Gǫ µσ Gα ǫν Gǫ µν Gα ǫσ + Gα µǫ T ǫ νσ. () P α νµσ = u γ α νσ G α νµ σ + (Gǫ νµ C ǫ νµ)γ α ǫσ (G α ǫµ C α ǫµ)γ ǫ νσ + P ǫ σµg α νǫ. Prf. We prve (a) and (c) nly. The prf f the ther parts s smlar. (a) We have (c) We have R α µσν = δ ν Γ α µσ δ σ Γ α µν + Γ ǫ µσ Γ α ǫν Γ ǫ µν Γ α ǫσ + C α µǫ Rǫ σν = δ ν Γ α σµ δ σ Γ α νµ + Γ ǫ σµγ α νǫ Γ ǫ νµγ α σǫ + C α ǫµr ǫ σν = {δ ν Γ α σµ + Γǫ σµ (Λα νǫ + Γα ǫν )} {δ σγ α νµ + Γǫ νµ (Λα σǫ + Γα ǫσ )} + C α ǫµ Rǫ σν = (δ ν Γ α σµ + Γ ǫ σµγ α ǫν) (δ σ Γ α νµ + Γ ǫ νµγ α ǫσ) (Γ ǫ σµλ α ǫν + Γ ǫ νµλ α σǫ) + C α ǫµr ǫ σν = (R α σµν Cα σǫ Rǫ µν + δ µγ α σν + Γǫ σν Γα ǫµ ) (Rα νµσ Cα νǫ Rǫ µσ + δ µ Γ α νσ + Γǫ νσ Γα ǫµ ) (Γǫ σµ Λα ǫν + Γǫ νµ Λα σǫ ) + Cα ǫµ Rǫ σν. = δ µ Λ α σν + Γ α ǫµλ ǫ σν Γ ǫ σµλ α ǫν Γ ǫ νµλ α σǫ + C α ǫµr ǫ σν + C α σǫr ǫ νµ + C α νǫr ǫ µσ = Λ α σν µ + Cα ǫµ Rǫ σν + Lα σνµ + Lα νµσ. P α νµσ = C α µν e σ µ Γ α σν ( µ N ǫ σ Γ ǫ σµ)c α ǫν = C α νµ σ + (C α µν e σ Cα νµ σ) µ Λ α σν µ Γ α νσ µ N ǫ σ(t α ǫν + C α νǫ) + (Λ ǫ σµ + Γǫ µσ )(T α ǫν + Cα νǫ ) = P α νµσ ( µ N ǫ σ Γǫ µσ )T α ǫν µ Λ α σν + Λǫ σµ Cα ǫν + (Cα µν e σ Cα νµ σ ) = (C α µν e σ Cα νµ σ) + Λ ǫ σµc α ǫν µ Λ α σν P ǫ σµt α ǫν = T α µν σ + Cǫ µν Λα σǫ Cα µǫ Λǫ σν µ Λ α σν P ǫ σµ T α ǫν = T α µν σ µ Λ α σν + (T ǫ µν + C ǫ νµ)λ α σǫ (T α µǫ + C α ǫµ)λ ǫ σν P ǫ σµt α ǫν = T α µν σ Λα σν µ + T ǫ µν Λα σǫ T α µǫ Λǫ σν Λα ǫν Cǫ σµ P ǫ σµ T α ǫν. 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

11 11 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 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 ξ µν := γ µν α α øξ µν := G µν α α γ µν := β α γ µν α øγ µν := B α G µν α η µν := β ǫ Λ ǫ µν φ µν := β ǫ Ω ǫ µν øη µν := B ǫ T ǫ µν øφ µν := B ǫ D ǫ µν χ µν := Λ α µν α ψ µν := Ω ǫ µν ǫ øχ µν := T α µν α øψ µν := D α µν α ǫ µν := 1 2 (β µ ν β ν µ ) θ µν := 1 2 (β µ ν + β ν µ ) øǫ µν := 1 2 (B µ ν B ν µ ) øθ µν := 1 2 (B µ ν + B ν µ ) k µν := γ ǫ αµ γα νǫ γǫ µα γα ǫν h µν := γ ǫ αµ γα νǫ + γǫ µα γα ǫν øk µν := G ǫ αµ Gα νǫ Gǫ µα Gα ǫν øh µν := G ǫ αµ Gα νǫ + Gǫ µα Gα ǫν σ µν := γ ǫ αµ γα ǫν øσ µν := G ǫ αµ Gα ǫν ω µν := γ ǫ µαγ α νǫ øω µν := G ǫ µαg α νǫ α µν := β µ β ν øα µν := B µ B ν 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:

12 12 Table 3: Summary f the fundamental scalars Hrzntal α := β µ β µ θ := β µ µ φ := β ǫ Ω ǫµ µ ψ := Ω αµ µ α ω := γ ǫµ α γ α µǫ σ := γ ǫ α µ γ α ǫµ h := 2γ αµ ǫ γ ǫ αµ Vertcal øα := B µ B µ øθ := B µ µ øφ := B ǫ D ǫµ µ øψ := D αµ µ α øω := G ǫµ α G α µǫ øσ := G ǫ α µ G α ǫµ øh := 2G αµ ǫ G ǫ 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. Table 4: Summary f the fundamental tensrs f zer trace Hrzntal Vertcal φ µν + 2α µν øφ µν + 2ᾱ µν ψ µν + 2θ µν øψ µν + 2øθ µν h µν + 2ω µν øh µν + 2øω µν 1 2 (φ µν ψ µν ) + θ µν α µν 1 2 g µνβ α 1 e α 2 (øφ µν øψ µν ) + øθ µν øα µν 1 2 g µνb α α e 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 αǫr ǫ µν, M µν := L α µαν = C α µǫ R ǫ αν, N µν := C α ǫµ R ǫ αν, F µν := C α ǫµ R ǫ αν. Then, clearly T µν := M µν N µν = T α µǫ R ǫ αν, G µν := M µν F µν = G α µǫ R ǫ αν, G µν T µν = G α ǫµ R ǫ 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.

13 13 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 expressns fr S σν, Ŝσν and S σν. Then, we have wth smlar (a) R σν = β σ ν β ν σ + β ǫ Λ ǫ σν + B ǫrσν ǫ, (b) S σν = B σ ν B ν σ + B ǫ Tσν ǫ, (c) R σν = 1 R 2 σν, (d) Ŝσν = 1 S 2 σν, (e) R σν = S σν = 0. Prpstn 6.2. Let R µσ := R µσα α, Rµσ := R µσα α and R µσ := R α µσα expressns fr S µσ, Ŝµσ and S µσ. Then, we have wth smlar (a) R µσ = β σ µ + CǫµR α σα ǫ + L α σαµ + L α αµσ, (b) S µσ = B σ µ, (c) R µσ = 1 R 2 µσ + 1{β 4 ǫλ ǫ σµ + Λǫ ασ Λα µǫ }, (d) Ŝµσ = 1 S 2 µσ + 1{B 4 ǫtσµ ǫ + T ασ ǫ T µǫ α }, (e) R µσ = β µ σ γµσ α α + β ǫγµσ ǫ γµǫγ α σα ǫ + G α µǫrασ, ǫ (f) S µσ := S α µσα = B µ σ G α µσ α + B ǫg ǫ µσ Gα µǫ Gǫ σα. Prpstn 6.3. The fllwng hlds. (a) R [µσ] = 1 {β 2 σ µ β µ σ } + CǫαR ǫ µσ α + C(ασ) ǫ Rα ǫµ C(αµ) ǫ Rα ǫσ, (b) R (µσ) = 1{β 2 σ µ + β µ σ + Tαµ ǫ Rα σǫ + T ασ ǫ Rα µǫ }, (c) S [µσ] = 1{B 2 σ µ B µ σ }, (d) S (µσ) = 1{B 2 σ µ + B µ σ }, (e) R [µσ] = 1 R 2 [µσ] + 1 β 4 ǫ Λ ǫ σµ, (f) R (µσ) = 1 R 2 (µσ) Λǫ ασ Λ α µǫ, (g) Ŝ[µσ] = 1 S 2 [µσ] + 1 B 4 ǫ Tσµ, ǫ

14 14 (h) Ŝ(µσ) = 1 2 S (µσ) T ǫ ασ T α µǫ, () R [µσ] = 1 2 {Lα αµσ + C α σǫ R ǫ αµ C α µǫ R ǫ ασ}, (j) R (µσ) = 1 2 {(β µ σ + β σ µ ) Ω α µσ α + β ǫ Ω ǫ µσ } γα µǫ γǫ σα {Gα µǫ Rǫ ασ + Gα σǫ Rǫ αµ }, (k) S [µσ] = 0, (l) S (µσ) = 1 2 {(B µ σ + B σ µ ) D α µσ α + B ǫ D ǫ µσ } Gα µǫ Gǫ σα. Crllary 6.4. The fllwng hlds: (a) R σ σ := g µσ Rµσ = β σ σ + T ǫσ α R α ǫσ, (b) S σ σ := gµσ Sµσ = B σ σ, (c) R σ σ := g µσ Rµσ = 1 2 {βσ σ + T ǫσ α R α ǫσ} Λǫσ α Λ α ǫσ, (d) Ŝσ σ := g µσ Ŝ µσ = 1 2 Bσ σ T ǫσ α T α ǫσ, (e) R σ σ := g µσ R µσ = β σ σ 1 2 Ωασ σ α β α Ω ασ σ γ ασ ǫ γ ǫ σα + Gασ ǫ R ǫ ασ, (f) S σ σ := g µσ S µσ = B σ σ 1 2 Dασ σ α B α D ασ σ G ασ ǫ G ǫ 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 ǫγ ǫ µσ + γǫ µα γα ǫσ + Gα µǫ Rǫ ασ. (b) S µσ = B µ σ G α µσ α B ǫg ǫ µσ + Gǫ µα Gα ǫσ. Prf. We prve (a) nly; the prf f (b) s smlar. We have 0 = R α µσα = (δ αγ α µσ δ σγ α µα ) + (Γǫ µσ Γα ǫα Γǫ µα Γα ǫσ ) + Rǫ σα Cα µǫ = δ α ( Γ α µσ + γα µσ ) δ σ( Γ α µα + γα µα ) + ( Γ ǫ µσ + γǫ µσ )( Γ α ǫα + γα ǫα ) Cnsequently, ( Γ ǫ µα + γǫ µα )( Γ α ǫσ + γα ǫσ ) + Rǫ σα Cα µǫ = R µσ (δ σ γµα α γǫα α Γ ǫ µσ) + (δ α γµσ α + γµσ ǫ Γ α ǫα γǫσ α γµǫ α Γ ǫ σα ) + Rǫ σα (Cα µǫ C α µǫ ) + γǫ µσ γα ǫα γǫ µα γα ǫσ. Γ ǫ µα R µσ = β µ σ γ α µσ α β ǫγ ǫ µσ + γǫ µα γα ǫσ + Gα µǫ Rǫ ασ. In vew f Prpstn 6.2 (e) and (f) and Prpstn 6.5, we btan

15 15 Crllary 6.6. The fllwng denttes hlds: (a) (β µ σ β µ σ ) (γ α µσ α γα µσ α) = (γǫ µαω α σǫ 2β ǫ γ ǫ µσ) (b) (B µ σ B µ σ ) (G α µσ α Gα µσ α) = (Gǫ µαd α σǫ 2B ǫ G ǫ µσ). The next tw tables summarze the results btaned n ths sectn, where the cntracted curvatures are expressed n terms f the fundamental tensrs. Table 5 (a): Secnd rank curvature tensrs Skew-symmetrc Symmetrc Dual R[µσ] = ǫ σµ L σµ + M [σµ] + N [σµ] R(µσ) = θ µσ + M (µσ) N (µσ) S [µσ] = øǫ σµ S(µσ) = øθ µσ Symmetrc R[µσ] = 1 2 R [µσ] η σµ R(µσ) = 1 2 R (µσ) {h µσ ω µσ σ µσ } Ŝ [µσ] = 1 2 S [µσ] øη σµ Ŝ (µσ) = 1 2 S (µσ) {øh µσ øω µσ øσ µσ } Remannan R [µσ] = 1 2 L µσ F [µσ] R(µσ) = θ µσ 1 2 (ψ µσ φ µσ ) ω µσ + M (µσ) F (µσ) S [µσ] = 0 S (µσ) = øθ µσ 1 2 (øψ µσ øφ µσ ) øω µσ Table 5 (b): h- and v-scalar curvature tensrs h-scalar curvature v-scalar curvature Dual Rσ σ = θ + T Sσ σ = øθ Symmetrc Rσ σ = 1(θ + T) 1(3ω + σ) 2 4 Ŝσ σ = 1øθ 1 (3øω + øσ) 2 4 Remannan R σ σ = θ 1 (ψ φ) ω + G 2 S σ σ = øθ 1 (øψ øφ) øω 2

16 16 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 = (Nβ α, Γα µν, Cα µν ), the hrzntal W-tensr (hw-tensr) Hα µνσ s defned by the frmula λ µ νσ λ µ σν = λ ǫ Hµνσ, ǫ whereas the vertcal W-tensr (vw-tensr) V α µνσ λ µ νσ λ µ σν = λ ǫ V ǫ µνσ, s defned by the frmula where and are the hrzntal and the vertcal cvarant dervatves wth respect t the cnnectn D. 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 + Λǫ νσλ α µǫ + S µ,ν,σ L α µσν. (b) Ṽ α µνσ = T α σν µ + T ǫ νσ T α µǫ. (c) Ĥα µνσ = 1 2 (Λα µν σ Λα µσ ν ) (Λǫ µν Λα σǫ Λǫ µσ Λα νǫ ) (Λǫ σν Λα ǫµ ). (d) H α µνσ = γα µν σ γα µσ ν + γǫ µσ γα ǫν γǫ µν γα ǫσ + Λǫ νσ γα µǫ. Prf. We prve (a) nly. The prf f the ther parts s smlar. We have λ ǫ Hǫ µνσ = λ ǫ Rǫ µσν + λ µ e ǫ Λǫ σν + λ µ e ǫ R ǫ σν. Hence, takng nt accunt Therem 4.6 (a), we btan H µνσ α = R µσν α + λ α (δ ǫ λ µ λ β Γ β ǫµ ) Λ ǫ σν + λ α ( ǫ λ µ λ β C ǫµ β )Rǫ σν = R µσν α + Λǫ νσ (Γα µǫ Γα ǫµ ) + Rǫ σν (Cα µǫ Cα ǫµ ) = Λ α σν µ + C α ǫµr ǫ σν + L α σνµ + L α νµσ + Λ ǫ νσλ α µǫ + T α µǫr ǫ σν = Λ α σν µ + T α ǫµ Rǫ σν + Cα µǫ Rǫ σν + Lα σνµ + Lα νµσ + Λǫ νσ Λα µǫ + T α µǫ Rǫ σν = Λ α σν µ + Λǫ νσ Λα µǫ + S µ,ν,σl α µσν.

17 17 Prpstn 7.3. Let Hνσ := H ανσ α, Ĥ νσ := Ĥα ανσ and H νσ := H α ανσ wth smlar expressn fr Ṽνσ. Then, we have (a) H νσ = β σ ν β ν σ + 2β ǫ Λ ǫ σν, (b) Ṽνσ = B σ ν B ν σ + 2B ǫ Tσν, ǫ (c) Ĥνσ = 1 { H 2 νσ + β ǫ Λ ǫ νσ}, (d) H νσ = 0. Prpstn 7.4. Let H µσ := H µασ α, Ĥ µσ := Ĥα µασ and H µσ := H α µασ expressns fr Ṽµσ. Then, we have wth smlar (a) H µσ = β σ µ + Λ ǫ ασλ α µǫ + S α,µ,σ L α αµσ, (b) Ṽµσ = B σ µ + T ǫ ασ T α µǫ, (c) Ĥµσ = 1 2 H µσ (β ǫλ ǫ σµ + Λǫ σα Λα µǫ ), (d) H µσ = β µ σ γ α µσ α + β ǫγ ǫ µσ γǫ σα γα µǫ. 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 ǫ ασ Λα µǫ, (c) Ṽ[µσ] = 1 2 {B σ µ B µ σ }, (d) Ṽ(µσ) = 1 2 {B σ µ + B µ σ } + T ǫ ασt α µǫ, (e) Ĥ[µσ] = 1 2 H [µσ] β ǫλ ǫ σµ, (f) Ĥ(µσ) = 1 2 H (µσ) Λǫ σα Λα µǫ, (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 ǫω ǫ µσ } γα µǫ γǫ σα. Crllary 7.6. the fllwng hlds: (a) H α α = βα α + Λ ǫµ αλ α ǫµ, (b) Ṽ α α = Bα α + T ǫµ αt α ǫµ, (c) Ĥα α = 1 2 βα α Λǫµ αλ α ǫµ, (d) H σ σ = β σ σ 1 2 Ωασ σ α β αω ασ σ γ ασ ǫγ ǫ σα. Takng nt accunt Prpstn 4.4, Therem 7.2 and the Banch dentty [4] fr the Remannan d-cnnectn, we get the fllwng

18 18 Prpstn 7.7. The hw-tensrs H α µνσ, Ĥα µνσ, H α µνσ and the vw-tensrs Ṽ α µνσ satsfy the fllwng denttes: (a) S µ,ν,σ Hα µνσ = 2S µ,ν,σ (Λ α µǫ Λǫ νσ + Lα µσν ). (b) S µ,ν,σ Ṽ α µνσ = 2S µ,ν,σ(t α µǫ T ǫ νσ ). (c) S µ,ν,σ Ĥ α µνσ = S µ,ν,σ L α µσν. (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] = ǫ σµ L σµ + 2M [σµ] H(µσ) = θ µσ (ω µσ + σ µσ h µσ ) Ṽ [µσ] = ǫ σµ Ṽ (µσ) = θ µσ ( ω µσ + σ µσ h µσ ) Symmetrc Ĥ [µσ] = 1 2 H [µσ] η σµ Ĥ (µσ) = 1 2 H (µσ) {ω µσ + σ µσ h µσ } 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 = 1θ 1 (3ω + σ) 2 4 Remannan H σ σ = θ 1 (ψ φ) ω 2

19 19 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 nn-lnear 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)). 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. We are aware that the present paper s f cmputatnal nature. The paper s certanly 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.

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