Extended Absolute Parallelism Geometry

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1 Extended Absolute Parallelsm Geometry arxv: v4 [math.dg] 5 Aug 2009 Nabl. L. Youssef and A. M. Sd-Ahmed Department of Mathematcs, Faculty of Scence, Caro Unversty, Gza, Egypt nlyoussef2003@yahoo.fr, nyoussef@frcu.eun.eg amrsdahmed@gmal.com, amrs@maler.eun.eg Abstract. In ths paper, we study Absolute Parallelsm (AP-) geometry on the tangent bundle T M of a manfold M. Accordngly, all geometrc objects defned n ths geometry are not only functons of the postonal argument x, but also depend on the drectonal argument y. Moreover, many new geometrc objects, whch have no counterpart n the classcal AP-geometry, emerge n ths dfferent framework. We refer to such a geometry as an Extended Absolute Parallelsm (EAP-) geometry. The buldng blocks of the EAP-geometry are a nonlnear connecton assumed gven a pror and 2n lnearly ndependent vector felds (of specal form) defned globally on T M defnng the parallelzaton. Four dfferent d-connectons are used to explore the propertes of ths geometry. Smple and compact formulae for the curvature tensors and the W-tensors of the four defned d-connectons are obtaned, expressed n terms of the torson and the contorton tensors of the EAP-space. Further condtons are mposed on the canoncal d-connecton assumng that t s of Cartan type (resp. Berwald type). Important consequences of these assumptons are nvestgated. Fnally, a specal form of the canoncal d-connecton s studed under whch the classcal AP-geometry s recovered naturally from the EAP-geometry. Physcal aspects of some of the geometrc objects nvestgated are ponted out and possble physcal mplcatons of the EAP-space are dscussed, ncludng an outlne of a generalzed feld theory on the tangent bundle TM of M. 1 Keywords: Parallelzable manfold, Absolute Parallelsm, Extended Absolute Parallelsm, Metrc d-connecton, Canoncal d-connecton, W-tensor, Feld equatons, Cartan type, Berwald type AMS Subject Classfcaton. 53B40, 53A40, 53B50. 1 ArXv number:

2 0. Introducton The geometry of parallelzable manfolds or the Absolute Parallelsm geometry (APgeometry) ([4], [11], [12], [14], [17]) has many advantages n comparson to Remannan geometry. Unlke Remannan geometry, whch has ten degrees of freedom (correspondng to the metrc components for n = 4), AP-geometry has sxteen degrees of freedom (correspondng to the number of components of the four vector felds defnng the parallelzaton). Ths makes AP-geometry a potental canddate for descrbng physcal phenomena other than gravty. Moreover, as opposed to Remannan geometry, whch admts only one symmetrc lnear connecton, AP-geometry admts at least four natural (bult-n) lnear connectons, two of whch are non-symmetrc and three of whch have non-vanshng curvature tensors. Last, but not least, assocated wth an AP-space there s a Remannan structure defned n a natural way. Thus, AP-geometry contans wthn ts geometrc structure all the mathematcal machnery of Remannan geometry. Accordngly, a comparson can be made between the results obtaned n the context of AP-geometry and general relatvty, whch s based on Remannan geometry. Thegeometryofthetangentbundle(TM, π, M)ofasmoothmanfoldM sveryrch. It contans a lot of geometrc objects of theoretcal nterest and of a great mportance n the constructon of varous geometrc models whch have proved very useful n dfferent physcal theores. Examples of such theores are the general theory of relatvty, partcle physcs, relatvstc optcs and others. In ths paper, we study AP-geometry n a context dfferent from the classcal one. Instead of dealng wth geometrc objects defned on the manfold M, as n the case of classcal AP-space, we are dealng wth geometrc objects defned on the tangent bundle T M of M. Accordngly, all geometrc objects consdered are, n general, not only functons of the postonal argument x, but also depend on the drectonal argument y. The paper s organzed n the followng manner. In secton 1, followng the ntroducton, we gve a bref account of the basc concepts and defntons that wll be needed n the sequel. The defntons of a d-connecton, d-tensor feld, torson, curvature, hv-metrc and metrc d-connecton on T M are recalled. We end ths secton by the constructon of a (unque) metrc d-connecton on TM whch we refer to as the natural metrc d-connecton. In secton 2, we ntroduce the Extended Absolute Parallelsm (EAP-) geometry by assumng that T M s parallelzable [3] and equpped wth a nonlnear connecton. The canoncal d-connecton s then defned, expressed n terms of the natural metrc d-connecton. In analogy to the classcal AP-geometry, two other d-connectons are ntroduced: the dual and the symmetrc d-connectons. We end ths part wth a comparson between the classcal AP-geometry and the EAP-geometry. In secton 3, we carry out the task of computng the dfferent curvature tensors of the four defned d-connectons. They are expressed, n a relatvely compact form, n terms of the torson and the contorton tensors of the EAP-space. All admssable contractons of these curvature tensors are also obtaned. In secton 4, we ntroduce and nvestgate the dfferent W-tensors correspondng to the dfferent d-connectons defned n the EAPspace, whch are agan expressed n terms of the torson and the contorton tensors. In sectons 5 and 6, we assume that the canoncal d-connecton s of Cartan and Berwald type respectvely. Some nterestng results are obtaned, the most mportant of whch s 2

3 that, n the Cartan type case, the gven nonlnear connecton s not ndependent of the vector felds formng the parallelzaton, but can be expressed n terms of ther vertcal counterparts. In secton 7, we further assume that the canoncal d-connecton s both of Cartan and Berwald type. We show that, under ths assumpton, the classcal APgeometry s recovered, n a natural way, from the EAP-geometry. In secton 8, we end ths paper wth some concludng remarks whch reveal possble physcal applcatons of the EAP-space; among them s an outlne of a generalzed feld theory on the tangent bundle T M of M, based on Euler-Lagrange equatons [8] appled to a sutable scalar Lagrangan. 1. Fundamental Prelmnares In ths secton we gve a bref account of the basc concepts and defntons that wll be needed n the sequel. Most of the materal covered here may be found n [8], [9] wth some slght modfcatons. Let M be a paracompact manfold of dmenson n of class C. Let π : TM M be ts tangent bundle. If (U, x µ ) s a local chart on M, then (π 1 (U),(x µ, y a )) s the correspondng local chart on TM. The coordnate transformaton on TM s gven by: x µ = x µ (x ν ), y a = p a a y a, a = ya = xa y a µ = 1,...,n; a = 1,...,n; p a and det(p a x a a ) 0. The paracompactness of M ensures the exstence of a nonlnear connecton N on TM wth coeffcents Nα a(x,y). The transformaton formula for the coeffcents Nα a s gven by N a α = pa a p α α Na α +p a a p a c α yc, (1.1) where p a c α = pa c x α. The nonlnear connecton leads to the drect sum decomposton T u (TM) = H u (TM) V u (TM), u TM \{0}. (1.2) Here, V u (TM) s the vertcal space at u wth local bass a := y a, whereas H u (TM) s thehorzontal space atuassocatedwth N supplementary tothevertcal spacev u (TM). The canoncal bass of H u (TM) s gven by δ µ := µ N a µ a, (1.3) where µ :=. Now, let (dx α, δy a ) be the bass of T x u(tm) dual to the adapted bass µ (δ α, a ) of T u (TM). Then δy a := dy a +Nα a dxα (1.4) and dx α (δ β ) = δ α β, dxα ( a ) = 0; δy a (δ β ) = 0, δy a ( b ) = δ a b. (1.5) Any vector feld X X(TM) s unquely decomposed n the form X = hx + vx, where h and v are respectvely the horzontal and the vertcal projectors assocated wth the decomposton (1.2). In the adapted frame (δ ν, a ), hx = X α δ α and vx = X a a. 3

4 Defnton 1.1. A nonlnear connecton N a µ s sad to be homogeneous f t s postvely homogeneous of degree 1 n the drectonal argument y. Defnton 1.2. A d-connecton on TM s a lnear connecton on TM whch preserves by parallelsm the horzontal and vertcal dstrbuton: f Y s a horzontal (vertcal) vector feld, then D X Y s a horzontal (vertcal) vector feld, for all X X(TM). Consequently, a d-connecton D on T M has only four coeffcents. The coeffcents of a d-connecton D = (Γ α µν, Γ a bν, Cα µc, Cbc a ) are defned by D δν δ µ =: Γ α µνδ α, D δν b =: Γ a bν a ; D c δ µ =: Cµcδ α α, D c b =: Cbc a a. (1.6) The transformaton formulae of a d-connecton are gven by: Γ α µ ν = pα α pµ µ p ν ν Γα µν +pα ǫ pǫ µ ν, Γa b µ = pa a pb b pµ µ Γ a bµ +pa c pc b µ ; C α µ c = pα α pµ µ p c c Cα µc, Ca b c = pa a pb b pc c Ca bc. A comment on notaton: Both Greek ndces {α, β, µ,...} and Latn ndces {a,b,c,...},asprevouslymentoned, takevaluesfromthesameset{1,...,n}. Itshouldbe noted, however, that Greek ndces are used to denote horzontal counterpart, whereas Latn ndces are used to denote vertcal counterpart. Ensten conventon s appled on both types of ndces. Defnton 1.3. A d-tensor feld T on TM of type (p,r; q,s) s a tensor feld on TM whch can be locally expressed n the form where u {α, a }, v j {β j, b j }, T = T u 1...u p+r v 1...v q+s u1... up+r dx v 1... dx v q+s, u {δ α, a }, dx v j {dx β j, δy b j }, = 1,...,p+r; j = 1,...,q+s, so that the number of α s = p, the number of a s = r, the number of β j s = q and the number of b j s = s. Let T = Tβb αaδ α a dx β δy b be a d-tensor feld of type (1,1;1,1). Let X X(TM) be such that X = hx +vx = X µ δ µ +X c c. Then, by the propertes of a d-connecton, we have D h XT := D hx T = (X µ T αa βb µ)δ α a dx β δy b, where Smlarly, where T αa βb µ := δ µ T αa βb +Tǫa βb Γα ǫµ +Tαd βb Γa dµ Tαa ǫb Γǫ βµ Tαa βd Γd bµ. (1.7) D v XT := D vx T = (X c T αa βb c)δ α a dx β δy b, T αa βb c := c T αa βb +Tǫa βb Cα ǫc +Tαd βb Ca dc Tαa ǫb Cǫ βc Tαa βd Cd bc. (1.8) It s evdent that (1.7) and (1.8) can be wrtten for any d-tensor feld of arbtrary type. 4

5 Defnton 1.4. The two operators D h X (denoted locally by ) and D v X (denoted locally by ) are called respectvely the horzontal (h-) and vertcal (v-) covarant dervatves assocated wth the d-connecton D. Defnton 1.5. The torson T of a d-connecton D on TM s defned by where T(X,Y) := D X Y D Y X [X,Y]; X,Y X(TM). (1.9) For gettng the local expresson for T, we frst recall that [δ µ, δ ν ] = R a µν a ; [δ µ, b ] = ( b N a µ ) a, s the curvature of the nonlnear connecton. By a drect substtuton n formula (1.9), we obtan R a µν := δ νn a µ δ µn a ν (1.10) Proposton 1.6. In the adapted bass (δ α, a ), the torson tensor T of a d-connecton D = (Γ α µν, Γa bµ, Cα µc, Ca bc ) s charaterzed by the followng d-tensor felds wth the local coeffcents (Λ α µν, Rµν, a Cµc, α Pµc, a Tbc a ) defned by: ht(δ ν, δ µ ) =: Λ α µνδ α, vt(δ ν, δ µ ) =: R a µν a where ht( c, δ µ ) =: C α µcδ α, Λ α µν := Γα µν Γα νµ, vt( c, δ µ ) =: P a µc a, vt( c, b ) =: T a bc a, Pa µc := c N a µ Γa cµ, Ta bc := Ca bc Ca cb. (1.11) Throughout the paper we shall use the notaton T = (Λ α µν, R a µν, C a µc, P a µc, T a bc ). Corollary 1.7. The torson tensor T = (Λ α µν, Ra µν, Ca µc, Pa µc, Ta bc ) of a d-connecton D vanshes f Γ α µν = Γ α νµ, Rµν a = Cµc α = 0, c Nµ a = Γ a cµ, Cbc a = Ccb. a Defnton 1.8. The curvature tensor R of a d-connecton D s gven by R(X,Y)Z := D X D Y Z D Y D X Z D [X, Y] Z; X,Y,Z X(TM). By defnton of a d-connecton, t follows that R(X,Y)Z s determned by eght d- tensor felds, sx of whch are ndependent due to the fact that R(X,Y) = R(Y,X). We set R(δ µ, δ ν )δ β =: Rβµν α δ α; R(δ µ, δ ν ) b =: Rbµν a a, R( c, δ ν )δ β =: P α βνcδ α ; R( b, c )δ β =: S α βbc δ α; R( c, δ ν ) b =: P a bνc a, R( c, d ) b =: S a bcd a. ThroughoutthepaperweshallusethenotatonR = (R α βµν,ra bµν,pα βνc,pa bνc,sα βbc,sa bcd ). 5

6 Theorem 1.9. The curvature R of a d-connecton D = (Γ α µν, Γ a bµ, Cα µc, Cbc a ) s charaterzed by the d-tensor felds wth local coeffcents: (a) R α βµν = δ µγ α βν δ νγ α βµ +Γǫ βν Γα ǫµ Γ ǫ βµ Γα ǫν +C α βd Rd νµ, (b) R a bµν = δ µγ a bν δ νγ a bµ +Γc bν Γa cµ Γc bµ Γa cν +Ca bd Rd νµ, (c) P α βνc = c Γ α βν Cα βc ν +Cα βd Pd νc, (d) P a bνc = c Γ a bν Ca bc ν +Ca bd Pd νc, (e) S α βbc = b C α βc c C α βb +Cǫ βc Cα ǫb Cǫ βb Cα ǫc, (f) S a bcd = c C a bd d C a bc +Ce bd Ca ec Ce bc Ca ed. Corollary The curvature tensor R = (Rβµν α,ra bµν,pα βνc,pa bνc,sα βbc,sa bcd ) of a d- connecton D vanshes ff R α βµν = Ra bµν = Pα βνc = Pa bνc = Sα βbc = Sa bcd = 0. Defnton An hv-metrc on TM s a covarant d-tensor feld G := hg+vg on TM, where hg := g αβ dx α dx β, vg := g ab δy a δy b such that: g αβ = g βα, det(g αβ ) 0; g ab = g ba, det(g ab ) 0. (1.12) The nverses of (g αβ ) and (g ab ), denoted by (g αβ ) and (g ab ) repectvely, are gven by g αǫ g ǫβ = δ α β, g aeg eb = δ a b. (1.13) Defnton A d-connecton D on TM s sad to be metrc or compatble wth the metrc G f D X G = 0, X X(TM). In the adapted frame (δ α, a ), the above condton can be expressed locally n the form: g αβ µ = g αβ c = g ab µ = g ab c = 0. (1.14) We have the followng Theorem [8]: Theorem There exsts a unque metrcal d-connecton D = ( Γ α µν, Γ a bν, on TM wth the propertes that C α µc, C a bc ) (a) Λ α µν = Γ α µν Γ α νµ = 0, T a bc = C a bc C a cb = 0. (b) Γ a bν := b N a ν gac (δ ν g bc g dc b N d ν g bd c N d ν ), C α µc := 1 2 gαǫ c g µǫ. In ths case, the coeffcents Γ α µν and C a bc are necessarly of the form Γ α µν := 1 2 gαǫ (δ µ g ǫν +δ ν g ǫµ δ ǫ g µν ), Cbc a := 1 2 gad ( b g dc + c g db d g bc ). 6

7 Defnton The d-connecton D = ( Γ α µν, Γ a bν, C α µc, C a bc) defned n Theorem 1.13 wll be referred to as the natural metrc d-connecton. The h- and v-covarant dervatves wth respect to the natural metrc d-connecton D wll be denoted by o and o respectvely. 2. Extended Absolute Parallelsm Geometry (EAP-geometry) In ths secton, we study AP-geometry n a context dfferent from the classcal one. Instead of dealng wth geometrc objects defned on the manfold M, we wll be dealng wth geometrc objects defned on the tangent bundle T M of M. Many new geometrc objects, whch have no counterpart n the classcal AP-geometry, emerge n ths dfferent framework. Moreover, the basc geometrc objects of the new geometry acqure a rcher structure compared to the correspondng basc geometrc objects of the classcal APgeometry (See Table 2). As n the prevous secton, M s assumed to be a smooth paracompact manfold of dmenson n. Ths nsures the exstence of a nonlnear connecton on TM so that the decomposton (1.2) nduced by the nonlnear connecton holds. We assume that λ, = 1,...,n, are n vector felds defned globally on TM. In the adapted bass (δ α, a ), we have λ = hλ + vλ = λ α δ α + λ a a. We further assume that the n horzontal vector felds hλ and the n vertcal vector felds vλ are lnearly ndependent. Ths mples, n partcular, that the n vector felds λ, themselves, are lnearly ndependent. Moreover, we have λ α λ β = δβ α, λ α λ α = δ j ; j λ a λ b = δb a, λ a λ a = δ j, (2.1) j where ( λ α ) and ( λ a ) denote the nverse matrces of ( λ α ) and ( λ a ) respectvely. We refer to the above space, whch we denote by (TM, λ), as an Extended Absolute Parellelsm (EAP-) geometry whch s characterzed by the exstence of 2n lnearly ndependent vector felds defned globally on T M. The Latn ndces {,j} wll be used for numberng the n vector felds (mesh ndces). Ensten conventon s appled on the mesh ndces (whch wll always be wrtten n lower poston) as well as the component ndces. In the sequel, to smplfy notatons, we wll use the symbol λ wthout the subscrpt to denote any one of the vector felds λ ( = 1,...,n). The ndex wll appear only when summaton s performed. Let us defne Then, clearly, g αβ := λ α λ β, g ab := λ a λ b. (2.2) G = g αβ dx α dx β +g ab δy a δy b s an hv-metrc on TM. Moreover, n vew of (2.1), the nverse of the matrces (g αβ ) and (g ab ) are gven by (g αβ ) and (g ab ) respectvely, where g αβ = λ α λβ, g ab = λ a λb. (2.3) 7

8 Now, let D = ( Γ α µν, Γ a bν, C α µc, C a bc) be the natural metrc d-connecton defned by Theorem 1.13, where g µν and g ab are the metrc tensors gven by (2.2). Theorem 2.1. There exsts a unque d-connecton D = (Γ α µν, Γa bν, Cα µc, Ca bc ) such that λ α µ = λ α c = λ a µ = λ a c = 0, (2.4) where and are the h- and v-covarant dervatves wth respect to D. Consequently D s a metrc d-connecton. It s gven by Γ α µν := Γ α µν + λ α λ µ o ν, Γ a bν := Γ a bν + λ a λ b o ν ; (2.5) C α µc := C α µc + λ α λ µ o c, C a bc := C a bc + λ a λ b o c. (2.6) Relaton (2.4) wll be called the AP-condton (as n the classcal AP-geometry). Proof. Frst, t s clear that D s a d-connecton on TM. We next prove that λ α ν = 0. We have λ α ν = δ ν λ α +λ µ Γ α µν = δ νλ α + = (δ ν λ α + Γ α µνλ µ ) ( The rest s proved n a smlar manner. λ µ j λ µ ) λ µ ( Γ α µν + λ α λ j j µ o ν ) λ α o j ν = λ α o ν λ α o ν = 0. Defnton 2.2. The d-connecton D = (Γ α µν, Γ a bν, Cα µc, Cbc a ) defned n Theorem 2.1 wll be referred to as the canoncal d-connecton of the EAP-space. In analogy to the classcal AP-space, the torson tensor of the canoncal d-connecton wll be called the torson tensor of the EAP-space. Theorem 2.3. The canoncal d-connecton D can be expressed explctely n terms of the λ s only n the form: Γ α µν = λ α (δ ν λ µ ), Cµc α = λ α ( c λ µ ), Γ a bν = λ a (δ ν λ b ); (2.7) Cbc a = λ a ( c λ b ). (2.8) Proof. Snce λ α ν = 0, t follows that δ ν λ α = λ ǫ Γ α ǫν. Multplyng by λ µ, we get λ µ (δ ν λ α ) = Γ α µν so that, by (2.1), Γ α µν = λ α (δ ν λ µ ). The other formulae are derved n a smlar manner. By Theorem 2.1 and Theorem 2.3, we have Corollary 2.4. The natural metrc d-connecton D can be expressed explctely n terms of the λ s only n the form Γ α µν = λ α (δ ν λ µ λ µ o ν ), C α µc = λ α ( c λ µ λ µ o c ), Γ a bν = λ a (δ ν λ b λ b o ν ); (2.9) C a bc = λ a ( c λ b λ b o c ). (2.10) 8

9 Defnton 2.5. The contorton tensor of an EAP-space s defned by C(X, Y) := D Y X D Y X; X,Y X(TM), where D s the cannoncal d-connecton and D s the natural metrc d-connecton. In the adapted bass (δ µ, a ), the contorton tensor s characterzed by the followng d-tensor felds: C(δ µ, δ ν ) =: γ α µν δ α, C(δ µ, c ) =: γ α µc δ α; C( b, δ µ ) =: γ a bµ a, C( b, c ) =: γ a bc a ; γ α µν := Γα µν Γ α µν, γa bµ := Γa bµ Γ a bµ ; γα µc := Cα µc C α µc, γa bc := Ca bc C a bc. (2.11) Throughout the paper we shall use the notaton C = (γ α µν, γ a bµ, γα µc, γ a bc ). By defnton of the canoncal d-connecton and (2.11), the contorton tensor can be expressed explctely n terms of the λ s only n the form: γ α µν = λ α λ µ o ν, γ a bµ = λ a λ b o µ, γ α µc = λ α λ µ o c, γ a bc = λ a λ b o c. (2.12) Proposton 2.6. Let γ αµν := g αǫ γ ǫ µν, γ abµ := g ac γ c bµ, γ αµc := g αǫ γ ǫ µc, γ abc := g ad γ d bc. Then each of the above defned d-tensor felds s skew-symmetrc n the frst par of ndces. Consequently, γ α αν = γa aµ = γα αc = γa ac = 0. Proof. We have γ αµν +γ µαν = The rest s proved analogously. A smple calculaton gves λ α λ µ o ν + λ µ λ α o ν = ( λ α λ µ ) o ν = g αµ o ν = 0. Proposton 2.7. Let T = (Λ α µν, Ra µν, Cα µc, Pa µb, Ta bc ) and C = (γα µν, γa bµ, γα µc, γa bc ) be the torson and the contorton tensors of the EAP-space respectvely. Then the followng relatons hold: Λ α µν = γ α µν γ α νµ, P a µb = γ a bµ + P a µb, C α µc = γ α µc + C α µc, T a bc = γ a bc γ a cb. (2.13) Consequently, Λ α µα = γ α µα =: C µ, T a ba = γ a ba =: C b. (2.14) Remark 2.8. It can be shown, n analogy to the classcal AP-space [2], that γ αµν = 1 2 (Λ αµν +Λ νµα +Λ µνα ), γ abc = 1 2 (T abc +T cba +T bca ); (2.15) where Λ αµν := g αǫ Λ ǫ µν and T abc := g ad T d bc. By (2.13) and (2.15), Λ α µν (resp. Ta bc ) vanshes ff γα µν (resp. γa bc ) vanshes. Defnton 2.9. Let D = (Γ α µν, Γ a bµ, Cα µc, Cbc a ) be the canoncal d-connecton. 9

10 (a) The dual d-connecton D = ( Γ α µν, Γa bµ, Cα µc, Ca bc ) s defned by Γ α µν := Γ α νµ, Γa bµ := Γ a bµ; Cα µc := C α µc, Ca bc := C a cb. (2.16) (b) The symmertc d-connecton D = ( Γ α µν, Γa bµ, Ĉµc α, Ĉa bc ) s defned by Γ α µν := 1 2 (Γα µν +Γα νµ ), Γa bµ := Γ a bµ ; Ĉα µc := Cα µc, Ĉa bc := 1 2 (Ca bc +Ca cb ). (2.17) We shall denote the horzontal (vertcal) covarant dervatve of D and D by ( ) and ( ) respectvely. It follows mmedately from the above defnton that λ α e c = λ α b c = λ α c = 0, λ a e µ = λ a b µ = λ a µ = 0; (2.18) λ α e µ = λ β Λ α µβ, λα b µ = 1 2 λα e µ ; λ a e c = λ b Tcb a, λa b c = 1 2 λa e c. (2.19) As easly checked, we also have Proposton The covarant dervatves of the metrc G wth respect to the dual and symmetrc d-connectons D and D are gven respectvely by: g αβ e µ = Λ αβµ +Λ βαµ, g αβ e c = g ab e µ = 0, g ab e c = T abc +T bac ; (2.20) g αβ b µ = 1 2 g αβ e µ, g αβ b c = g ab b µ = 0, g ab b c = 1 2 g ab e c. (2.21) Consequently, D and D are non-metrc connectons. We end ths secton wth the followng tables. Table 1: Fundamental connectons of the EAP-space Connecton Coeffcents Covarant Torson Metrcty dervatve Natural ( Γ α µν, Γ a bν, C α µc, C a bc) (0,R a µν, C α µc, P a µc,0) metrc Canoncal (Γ α µν,γa bν,cα µc,ca bc ) (Λα µν,ra µν,cα µc,pa µc,ta bc ) metrc Dual (Γ α νµ,γa bν,cα µc,ca cb ) ( Λ α µν,ra µν,cα µc,pa µc, Ta bc ) non-metrc Symmetrc (Γ α (µν),γa bν,cα µc,ca (bc) ) (0,Rµν a,cα µc,pa µc,0) non-metrc The next table gves a comparson between the classcal AP-space and the EAP-space. We shall refer to the Remannan connecton n the classcal AP-space and the natural metrc d-connecton n the EAP-space smply as the metrc connecton. Moreover, we consder only the metrc and the canoncal connectons n both spaces. We also set L a bν := 1 2 gac (δ ν g bc g dc b Nν d g bd c Nν d). 10

11 Table 2: Comparson between classcal AP-geometry and EAP-geometry Classcal AP-geometry EAP-geometry Underlyng space M T M Buldng blocks λ α (x) N a µ(x,y), λ(x,y) = (λ α (x,y), λ a (x,y)) Metrc g µν = λ µ λ ν G = (g µν, g ab ); g µν = λ µ λ ν, g ab = λ a λ b Metrc connecton Γ α µν = 1 2 gαǫ ( µ g νǫ + ν g µǫ ǫ g µν ) D = ( Γ α µν, Γ a bν, C α µc, C a bc); Γ α µν = 1 2 gαǫ (δ µ g νǫ +δ ν g µǫ δ ǫ g µν ), Γ a bν = b N a ν +L a bν ; C α µc = 1 2 gαǫ c g µǫ, C a bc = 1 2 gad ( b g cd + c g bd d g bc ) Canoncal Γ α µν = λ α ( ν λ µ ) D = (Γ α µν, Γa bν, Cα µc, Ca bc ); connecton Γ α µν = λ α (δ ν λ µ ); Γ a bν = λa (δ ν λ b ), Cµc α = λ α ( c λ µ ); Cbc a = λ a ( c λ b ) AP-condton λ α µ = 0 λ α µ = λ α c = 0, λ a µ = λ a c = 0 Torson Λ α µν = Γ α µν Γ α νµ T = (Λ α µν, R a µν, C α µc, P a µc, T a bc ); Λ α µν = Γα µν Γα νµ ; Ra µν = δ νn a µ δ µn a ν, P a µc = c N a µ Γa cµ ; Ta bc = Ca bc Ca cb Contorson γ α µν = Γα µν Γ α µν C = (γ α µν, γa bν, γα µc, γa bc ); γ α µν = Γ α µν Γ α νµ; γ a bν = Γa bν Γ a bν, γ α µc = C α µc C α µc; γ a bc = Ca bc C a bc Basc vector C µ = Λ α µα = γα µα B = (C µ, C a ); C µ = Λ α µα = γ α µα; C a = T d ad = γd ad 11

12 3. Curvature tensors n the EAP-space Let (TM, λ) be an EAP space. Let D be the cannoncal d-connecton. Let D, D and D be the natural metrc d-connecton, the dual d-connecton and the symmetrc d- connecton respectvely. The curvature tensors of the four d-connectons wll be denoted respectvely by R, R, R and R. In ths secton, we carry out the task of calculatng the curvature tensors together wth ther contractons. Lemma 3.1. The followng commutaton formulae hold: (a) λ α ν µ λ α µ ν = R α βµν λβ Λ β νµλ α β R d νµλ α d (b) λ α ν c λ α c ν = P α βνc λβ C β νc λα β P d νc λα d (c) λ α b c λ α c b = S α βcb λβ T d bc λα d (d) λ a ν µ λ a µ ν = R a bµν λb Λ β νµλ a β R d νµλ a d (e) λ a ν c λ a c ν = P a bνc λb C β νc λa β P d νc λa d (f) λ a b c λ a c b = S a dcb λd T d bc λa d, In vew of (2.1), Corollary 1.10 and Theorem 2.1, Lemma 3.1 drectly mples that Theorem 3.2. The curvature tensor R = (Rβµν α,ra bµν,pα βνc,pa bνc,sα βbc,sa bcd ) of the canoncal d-connecton vanshes dentcally. Corollary 3.3. The followng denttes hold: Λ α βµ α = (C β µ C µ β )+C ǫ Λ ǫ βµ + S βµα R a µβc α αa (3.1) T d bc d = (C b c C c b )+C d T d bc. (3.2) Proof. The Banch denttes [9] appled to the canoncal d-connecton D gve and S β,µ,ν (Λ α βµ ν +Λǫ µν Λα βǫ +Ra βµ Cα νa ) = 0 (3.3) S b,c,d (Tbc d a +Te cd Ta be ) = 0, (3.4) where the notaton S β,µ,ν denotes a cyclc permutaton on the ndces β,µ,ν and summaton. (3.1) and (3.2) are obtaned by settng α = ν and a = d n (3.3) and (3.4) respectvely. By applyng the comutaton formula (c) of Lemma 3.1 wth respect to the dual and symmetrc d-connectons respectvely, takng nto account (2.1) and (2.18), we obtan S α βbc = Ŝα βbc = 0. Ths could be also deduced from Theorem 1.9 (e) and Theorem 3.2, notng that C α µc = C α µc = Ĉα µc. 12

13 In Theorems 3.4, 3.5 and 3.6 below, concernng the curvature tensors of the d- connectons D, D and D, we wll make use of Theorem 3.2, namely that the curvature tensors of the canoncal d-connecton vansh dentcally. Theorem 3.4. The curvature tensors of the natural metrc d-connecton D can be expressed n the form: (a) R α βµν = (γα βµ ν γα βν µ )+(γǫ βν γα ǫµ γǫ βµ γα ǫν ) γα βǫ Λǫ νµ γα βd Rd νµ, (b) R a bµν = (γa bµ ν γa bν µ )+(γd bν γa dµ γd bµ γa dν ) γa bǫ Λǫ νµ γa bd Rd νµ, (c) P α βνc = (γα βc ν γα βν c )+(γǫ βν γα ǫc γǫ βc γα ǫν ) γα βǫ Cǫ νc γα βd Pd νc, (d) P a bνc = (γa bc ν γa bν c )+(γd bν γa dc γd bc γa dν ) γa bǫ Cǫ νc γa bd Pd νc, (e) S α βbc = (γ α βb c γα βc b )+(γǫ βc γα ǫb γǫ βb γα ǫc) γ α βd Td cb, (f) S a bcd = (γ a bc d γa bd c )+(γe bd γa ec γ e bc γa ed ) γa be Te dc. Consequently, (g) R βµ := R α βµα = (γ α βµ α C β µ) C ǫ γ ǫ βµ +γα βǫ γǫ µα γ α βd Rd αµ, (h) R := g βµ R βµ = 1 2 (Ωαµ µ α C α Ω αµ µ) C µ µ +γ αµ ǫγ ǫ µα γ αµ dr d αµ, () P βc := P α βαc = (C β c γ α βc α )+C ǫγ ǫ βc +γα βǫ (Cǫ αc γǫ αc )+γα βd Pd αc, (j) P bν := P d bνd = (C b ν γ d bν d )+C dγ d bν γd be γe dν γd bǫ Cǫ νd γe bd Pd νe, (k) S bc := S d bcd = (γd bc d C b c) C d γ d bc +γd be γe cd, (l) S := g bc S bc = 1 2 (Ωad d a C a Ω ad d) C d d +γ ad cγ c da, where Ω α βµ := γα βµ +γα µβ, Ωa bc := γa bc +γa cb. Proof. We prove (a) and (c) only. The other formulae of the frst part are proved n a smlar manner. The second part s obtaned drectly by applyng the sutable contractons. (a) We have R α βµν = δ µ Γ α βν δ νγ α βµ + Γ ǫ βν Γ α ǫµ Γ ǫ βµ Γ α ǫν + C α βdrνµ d = δ µ (Γ α βν γα βν ) δ ν(γ α βµ γα βµ )+(Γǫ βν γǫ βν )(Γα ǫµ γα ǫµ ) (Γǫ βµ γǫ βµ ) (Γ α ǫν γα ǫν )+(Cα βd γα βd )Rd νµ = R α βµν +(δ νγ α βµ +γǫ βµ Γα ǫν γα ǫµ Γǫ βν ) (δ µγ α βν +γǫ βν Γα ǫµ γα ǫν Γǫ βµ ) γ α βd Rd νµ +(γǫ βν γα ǫµ γǫ βµ γα ǫν ) = (γ α βµ ν γα βν µ )+(γǫ βν γα ǫµ γǫ βµ γα ǫν ) γα βǫ Λǫ νµ γα βd Rd νµ 13

14 (c) We have P α βνc = cγ α βν C α βc o ν + C α βd P d νc = c Γ α βν c γ α βν C α βc ν +(C α βc ν C α βc o ν)+(cα βd γ α βd)(p d νc +γ d cν) = Pβνc α c γβν α +{Cα βc ν (Cα βc γα βc )o ν }+Cβd α γd cν γα βd γd cν γα βd Pd νc = c γβν α +(Cβcγ ǫ ǫν α Cǫcγ α βν ǫ Cβdγ α cν)+(γ d βc ν α γβcγ ǫ ǫν α +γǫcγ α βν ǫ + γ α βd γd cν )+Cα βd γd cν γα βd γd cν γα βd Pd νc = γ α βc ν ( c γ α βν +γǫ βν Cα ǫc γα ǫν Cǫ βc )+(γǫ βν γα ǫc γǫ βc γα ǫν ) γα βd Pd νc = (γ α βc ν γα βν c )+(γǫ βν γα ǫc γǫ βc γα ǫν ) γα βǫ Cǫ νc γα βd Pd νc Theorem 3.5. The non-vanshng curvature tensors of the dual d-connecton D can be expressed n the form: (a) R α βνµ = Λα µν β +S β,µ,νc α βa Ra µν, (b) R a bνµ = Rd µν Ta db, (c) P α βµc = Λα µβ c +Λα ǫβ Cǫ µc, (d) P a bµc = Ta bc µ +Ta db Pd µc, (e) S a bcd = Ta dc b. Consequently, (f) R βν := R α βνα = C ν β +S β,ν,α C α βa Ra αν, (g) R := g βµ Rβµ = C µ µ, (h) P βc := P α βαc = C β c +Λ α βǫ Cǫ αc, () P bµ := P a βµa = C b µ +T a db Pd µa, (j) S bd := S a bda = C d b, (k) S := g bd Sbd = C d d. Proof. (b) s a consequence of the commutaton formula (d) of Lemma 3.1 appled to the dual d-connecton, takng nto account (2.1), (2.18) and (2.19). (b) could be also obtaned fromtheorem 1.9 (b) and Theorem 3.2, notng that Γ a bµ = Γ a bµ and Ca bd = Ta bd + C bd a. We next prove (a) and (c) of the frst part. The second part follows mmedately by applyng the sutable contractons. 14

15 (a) We have (c) We have R α βνµ = δ ν Γα βµ δ µ Γα βν + Γ ǫ βµ Γ α ǫν Γ ǫ βν Γ α ǫµ + C α βar a µν = δ ν Γ α µβ δ µγ α νβ +Γǫ µβ Γα νǫ Γǫ νβ Γα µǫ +Cα βa Ra µν = {δ ν Γ α µβ +Γǫ µβ (Λα νǫ +Γα ǫν )} {δ µγ α νβ +Γǫ νβ (Λα µǫ + Γα ǫµ )}+Cα βa Ra µν = (R α µνβ C α µar a βν +δ β Γ α µν +Γ ǫ µνγ α ǫβ) (R α νµβ C α νar a βµ +δ β Γ α νµ + Γ ǫ νµγ α ǫβ) (Γ ǫ µβλ α ǫν +Γ ǫ νβλ α µǫ)+c α βar a µν = (δ β Λ α µν +Γα ǫβ Λǫ µν Γǫ µβ Λα ǫν Γǫ νβ Λα µǫ )+S β,µ,νc α βa Ra µν = Λ α µν β +S β,µ,νc α βa Ra µν P βµc α = c Γα βµ C α βc e + C α µ βd P µc d = c Γ α µβ Cα βc e µ +Cα βd Pd µc = ( c Γ α βµ Cα βc µ +Cα βd Pd µc )+ c Λ α µβ +(Cα βc µ Cα βc e ) µ = P α βµc +( c Λ α µβ +Λǫ µβ Cα ǫc Λα µǫ Cǫ βc ) = Λ α µβ c +Λ α ǫβc ǫ µc Theorem 3.6. The non-vanshng curvature tensors of the symmetc d-connecton D can be expressed n the form (a) R α βνµ = 1 2 (Λα βν µ Λα βµ ν )+ 1 4 (Λǫ βν Λα µǫ Λǫ βµ Λα νǫ )+ 1 2 (Λǫ νµ Λα βǫ ), (b) R a bνµ = 1 2 R a bνµ, (c) P α βµc = 1 2 P α βµc, (d) P a bµc = 1 2 P a bµc, (e) Ŝa bcd = 1 2 (Ta bc d Ta bd c )+ 1 4 (Te bc Ta de Te bd Ta ce )+ 1 2 (Te dc Ta eb ). Consequently, (f) R βν := R α βνα = 1 2 R βν 1 4 (C αλ α νβ +Λα νǫ Λǫ αβ ), (g) R := g βν Rβν = 1 2 R 1 4 Λαβ ǫλ ǫ αβ, (h) P βc := P α βαc = 1 2 P βc, () P bµ := P a βµa = 1 2 P bµ, (j) Ŝbd := Ŝa bda = 1 2 S bd 1 4 (C at a db +Ta de Te ab ), (k) Ŝ := gbd Ŝ bd = 1 2 S 1 4 Tab et e ab. Proof. Smlar to the proof of Theorems 3.4 and

16 4. Wanas tensors (W-tensors) The Wanas tensor, or smply the W-tensor, n the classcal AP-geometry, s a tensor whch measures the non-commutatvty of covarant dfferentatons of the parallelzaton vector feldsλ wth respect to the dual connecton: Wβνµ α := λ β ( λ α e ν e µ λ α e µ e ν ) (4.1) Ths tensor explctely contans the curvature and torson tensors. The W-tensor was frst defned by M. Wanas [12] and has been used by F. Mkhal and M. Wanas [5] to construct a geometrc theory unfyng gravty and electromagnetsm (GFT: generalzed feld theory). The scalar Lagrangan functon of the GFT s obtaned by double contractons of the tensor W αβ νµ. The symmetrc part of the feld equaton obtaned contans a second-order tensor representng the materal dstrbuton. Ths tensor s a pure geometrc, not a phenomologcal, object. The skew part of the feld equaton gves rse to Maxwell-lke equatons. The use of the W-tensor has thus aded to construct a geometrc theory va one sngle geometrc entty, whch Ensten was seekng for [1]. Varous sgnfcant applcatons (e.g [12], [13], [15]) have supported such a theory. Recently, the authors of ths paper nvestgated the most mportant propertes of ths tensor n the context of classcal AP-geometry [17]. The W-tensor was also studed by the present authors n the context of generalzed Lagrange spaces [16]. It should be noted that the W-tensor can be defned only n the context of APgeometry and ts generalzed versons (cf. e.g [16], [17]), snce t s defned only n terms of the vector felds λ s. Due to ts mportance n physcal applcatons, we are gong to nvestgate the propertes of the W-tensor n the present secton. The W-tensor (4.1) can be generalzed n the context of the EAP-geometry as follows. Defnton 4.1. Let(TM, λ) be an EAP-space. Foragven d-connectond = (Γ α µν, Γa bµ, Cµc α, Ca bc ), the W-tensor s gven by W = (W α βνµ, Wa bνµ, Wα βνc, Wa bνc, Wα βbc, Wa bcd ), (a) the hhh-tensor W α βνµ s defned by the formula λ α ν µ λ α µ ν = λ ǫ W α ǫνµ, (b) the hhv-tensor W a bνµ s defned by the formula λ a ν µ λ a µ ν = λ d W a dνµ, (c) the vhh-tensor W α βνc s defned by the formula λ α ν c λ α c ν = λ ǫ W α ǫνc, (d) the vhv-tensor W a bνc s defned by the formula λ a ν c λ a c ν = λ d W a dνc, 16

17 (e) the vvh-tensor Wβbc α s defned by the formula λ α b c λ α c b = λ ǫ W α ǫbc, (f) the vvv-tensor W a bcd s defned by the formula λ a c d λ a d c = λ e W a ecd, where and are the h- and v-covarant dervatves wth respect to the gven d-connecton D. Theorem 2.1, together wth (2.1), drectly mples that the W-tensors of the canoncal d-connecton vansh dentcally. In vew of Theorem 3.4, we obtan Theorem 4.2. The W-tensors correspondng to the natural metrc d-connecton D are gven by: (a) W α βνµ = (γα βµ ν γα βν µ )+(γǫ βν γα ǫµ γǫ βµ γα ǫν ) γα βǫ Λǫ νµ, (b) W a bνµ = (γ a bµ ν γa bν µ )+(γd bν γa dµ γd bµ γa dν ) γa bǫ Λǫ νµ, (c) W α βνc = (γ α βc ν γα βν c )+(γǫ βν γα ǫc γ ǫ βc γα ǫν)+(γ d cνγ α βd γǫ νcγ α βǫ ), (d) W a bνc = (γa bc ν γa bν c )+(γd bν γa dc γd bc γa dν )+(γd cν γa bd γǫ νc γa bǫ ), (e) W α βbc = S α βcb, (f) W a bcd = S a bdc. Proof. We prove (c) only. The other formulae are derved n a smlar manner. By defnton, we have λ ǫ W α ǫνc = λǫ P α ǫνc C ǫ νc λα o ǫ P d νc λα o d. Consequently, by (2.1), (2.12) and (2.13), we obtan W α βνc = P α βνc ( λ β λ α o ǫ )(Cνc ǫ γǫ νc ) ( λ β λ α o d )(Pνc d +γd cν ) = P α βνc +Cǫ νc γα βǫ +Pd νc γα βd (γǫ νc γα βǫ γd cν γα βd ) = (γ α βc ν γα βν c )+(γǫ βν γα ǫc γǫ βc γα ǫν )+(γd cν γα βd γǫ νc γα βǫ ) Snce λ a e µ = λ α e c = λ a b µ = λ α b c = 0, t follows, by defnton, that W a bνµ = W α βbc = Ŵa bνµ = Ŵα βbc = 0. Proceedng as n Theorem 4.2, takng nto account Theorem 3.5 and Theorem 3.6, we have the followng 17

18 Theorem 4.3. The non-vanshng W-tensors correspondng to the dual d-connecton D are gven by: (a) W α βνµ = Λα νµ β +Λǫ νµ Λα βǫ +S ν,µ,βc α βa Ra νµ, (b) W α βνc = Λα νβ c, (c) W a bνc = Ta bc ν, (d) W a bdc = Ta dc b +Te dc Ta be. Theorem 4.4. The non-vanshng W-tensors correspondng to the symmetrc d- connecton D are gven by: (a) Ŵα βνµ = R α βµν, (b) Ŵα βνc = 1 2 W α βνc, (c) Ŵa bνc = 1 2 W a bνc, (d) Ŵa bcd = Ŝa bdc. It s clear by the above theorem that the W-tensors correspondng to the symmetrc d-connecton gve no new d-tensor felds. Remark 4.5. The W-tensors correspondng to a gven d-connecton can be also defned covarantly n the form λ β µ ν λ β ν µ = λ ǫ W ǫ βµν, wth smlar expressons for the other counterparts. These expressons gve the same formulae (up to a sgn) for the W-tensors obtaned n Theorems 4.2, 4.3 and 4.4. Proposton 4.6. The followng denttes hold: (a) S β,µ,ν W α βνµ = S β,µ,ν Ŵ α βνµ = S β,µ,ν R a µβ Cα νa (b) S β,µ,ν Wα βνµ = 2S β,µ,ν R a µβ Cα νa Proof. By Theorem 4.2, we have S β,µ,ν W α βνµ = S β,µ,ν (γ α βµ ν γα βν µ )+S β,µ,ν(γ α βǫ Λǫ νµ )+S β,µ,ν (γ ǫ βν γα ǫµ γǫ βµ γα ǫν ) = S β,µ,ν (Λ α βµ ν +Λ ǫ µνλ α βǫ) = S β,µ,ν (Λ α βµ ν +Λǫ βµ Λα νǫ +Ra βµ Cα νa )+S β,µ,νr a µβ Cα νa = S β,µ,ν R a µβ Cα νa, where n the last step we have used (3.3). The proof of the other part of (a) s acheved by applyng the frst Banch dentty to the symmetrc d-connecton takng nto account Theorem 4.4 (a) together wth the fact that Ĉα νa = Cα νa. The proof of (b) s carred out n a smlar manner, agan by usng (3.3), takng nto consderaton Theorem 4.3 (a). 18

19 Summng up, the EAP-space has three dstnct W-tensors (correspondng to the natural metrc, dual and symmetrc d-connectons), each wth sx counterparts. Eght only out of the eghteen are ndependent, four concde wth the correspondng curvature tensors and four vansh dentcally. 5. Cartan-type case A drawback n the constructon of the EAP-space s the fact that the nonlnear connecton s assumed to exst a pror, ndependently of the vector felds λ s defnng the parallelzaton. It would be more natural and less arbtrary f the nonlnear connecton were expressed n terms of these vector felds. In ths case, all geometrc objects of the EAP-space wll be defned solely n terms of the buldng blocks of the space. Below, we mpose an extra condton on the canoncal d-connecton, namely, beng of Cartan type. The outcome of such a condton s the accomplshment of our requrment, besdes many other nterestng results. We frst recall the defnton of a Cartan type d-connecton [7], [8]. Defnton 5.1. A d-connectond = (Γ α µν, Γ a bµ, Cα µc, Cbc a ) on TM s sad to be of Cartan type f D h XC = 0; D v XC = vx; X X(TM), where C = y a a s the Louvlle vector feld. Locally, the above condtons are expressed n the form y a µ = 0, y a c = δ a c, (5.1) or, equvalently, Nµ a = yb Γ a bµ, yb Cbc a = 0. (5.2) Proposton 5.2. If a d-connecton D s of Cartan type, then the followng denttes nvolvng the torsons and curvatures hold: R a µν = yb R a bνµ, Pa µc = yb P a bµc, Ta bc = yd S a dcb. (5.3) Proof. Follows by applyng the commutaton formulae (d), (e) and (f) of Lemma 3.1 to the vector feld y a. Theorem 5.3. Let (T M, λ) be an EAP-space. Assume that the canoncal d-connecton D s of Cartan type. Then we have: (a) The expresson y b ( λ a µ λ b ) represents the coeffcents of a nonlnear connecton whch concdes wth the gven nonlnear connecton Nµ a : Na µ = yb ( λ a µ λ b ). 2 (b) R a µν = Pa µc = Ta bc = 0. Consequently, C a bc s symmetrc, γa bc = 0 and Ca bc = C a bc = Ĉa bc = C a bc. 2 A smlar expresson s found n ArXv: [gr-qc], but n a completely dfferent stuaton. 19

20 (c) λ a e b = λ a b b = λ a o b = 0, g ab e c = g ab b c = g ab o c = 0, y a e b = y a b b = y a o b = δ a b. (d) λ a are postvely homogeneous of degree 0 n y. Consequently, so are g ab. (e) b Nµ a = Γa bµ and Na µ s homogeneous. Consequently, Γ a bµ s postvely homogeneous of degree 0 n y. (f) γ a bµ = 0. Consequently, Γa bµ = Γ a bµ. (g) P a µb = 0. Proof. We have (a) Nµ a = yb ( λ a δ µ λ b ) = y b λ a ( µ N µ c c ) λ b = y b ( λ a µ λ b ) N c µ yb Cbc a = yb ( λ a µ λ b ). (b) s obtaned from Proposton 5.2, takng nto account Theorem 3.2. The vanshng of γ a bc s readly obtaned by Remark 2.8 and Ta bc = 0. (c) s a drect consequence of (b). (d) By the symmetry of Cbc a, we have 0 = yb Cbc a = yb Ccb a = λ a (y b b λ c ) so that, by (2.1), y b b λ c = 0. The result follows from Euler s Theorem. (e) follows from the fact that P a µb = b N a µ Γa bµ = 0 so that yb b N a µ = yb Γ a bµ = Na µ. Ths could be also deduced from the expresson obtaned for N a µ n (a), takng nto account that λ a (hence λ a ) are postvely homogeneous of degree 0 n y. (f) By (e), we have Γ a bµ = b Nµ a. Consequently, by defnton of the natural metrc d- connecton (Theorem 1.13), 1 2 gac (δ ν g bc g dc b Nν d g bd c Nν d) = Γ a bµ Γa bµ = γa bµ. Multplyng by g ae, we get 1 (δ 2 νg be g de b Nν d g bd e Nν d) = γ ebµ. Ths mples that γ ebµ s symmetrc n the ndces e,b. By Proposton 2.6, γ ebµ s also skewsymmetrc n thendces e,b. Consequently, γ ebµ vanshes so thatγbµ a = gae γ ebµ = 0. (g) s a drect consequence of the relaton P a µb = Pµb a + γa bµ, takng nto account that Pµb a = γa bµ = 0. Corollary 5.4. If the canoncal d-connecton s of Cartan type, then D, D and D are also of Cartan type. In what follows, we assume that the canoncal d-connecton s of Cartan type. The next two results are mmedate consequences of the fact that R a µν = Pa µc = Ta bc = γa bc = γa bµ = 0, takng nto consderaton Proposton 4.6 and Theorems 3.4, 3.5, 3.6, 4.2, 4.3 and 4.4. Proposton 5.5. The followng relatons hold: (a) R a bµν = P a bµc = S a bcd = 0, 20

21 (b) R βµν α = Λα νµ β, (c) R bµν a = R bµν a = P bµc a = P bµc a = S bcd a = Ŝa bcd = 0. (d) W α βνµ = R α βµν, (e) W a bµν = W a bµc = W a bcd = W bµc a = Ŵα bµc = W bcd a = Ŵa bcd = 0, (f) S β,µ,νw α βνµ = S β,µ,ν Ŵβνµ α = S W β,µ,ν βνµ α = 0. Consequently, W α βνµ, Ŵα βνµ and W βνµ α satsfy the frst Banch dentty of the Remannan curvature tensor. Theorem 5.6. The ndependent non-vanshng W-tensors are gven by: (a) W α βνc = (γα βc ν γα βν c )+(γǫ βν γα ǫc γǫ βc γα ǫν ) γǫ νc γα βǫ (b) W α βνµ = Λα νµ β +Λǫ νµ Λα βǫ, (c) W α βµc = Λα µβ c To sum up, the assumpton that the canoncal d-connecton beng of Cartan type mples that all the geometrc objects defned n the EAP-space are expressed n terms of the vector felds λ s only. The curvature of the nonlnear connecton vanshes and the three other defned d-connectons of the EAP-space, namely the dual, symmetrc and the natural metrc d-connectons, are also of Cartan type. Moreover, there are only seven non-vanshng curvature tensors and only three ndependent non-vanshng W-tensors, some of whch have smpler expressons than that obtaned n the general case. The EAP-space becomes rcher as new relatons among ts varous geometrc objects - whch are not vald n the general case - emerge. Accordngly, the EAP-space n ths case becomes more tangble, thus more sutable for physcal applcatons. We end ths secton wth the followng table. Table 3: EAP-geometry under the Cartan type case Connecton Coeffcents Torson Curvature Canoncal Dual (Γ α µν, b Nν a, Cα µc, Ca bc ) (Λα µν, 0, Cα µc, 0, 0) (0, 0, 0, 0, 0, 0) (Γ α νµ, b N a ν, Cα µc, Ca bc ) ( Λα µν, 0, Cα µc, 0, 0) ( R α βµν, 0, Pα βµc, 0, 0, 0) Symmetrc (Γ α (µν), b N a ν, Cα µc, Ca bc ) (0, 0, Cα µc, 0, 0) ( R α βµν, 0, Pα βµc, 0, 0, 0) Natural ( Γ α µν, b N a ν, C α µc, C a bc ) (0, 0, C α µc, 0, 0) ( R α βµν, 0, P α βµc, 0, S α βcd, 0) 21

22 6. Berwald-type case In ths secton we assume that the canoncal d-connecton D s of Berwald type [8], [10]. The consequences of ths assumpton are nvestgated. Defnton 6.1. A d-connecton D = (Γ α µν, Γ a bµ, Cα µc, Cbc a ) on TM s sad to be of Berwald type f b Nµ a = Γa bµ ; Cα µc = 0. (6.1) Proposton 6.2. If the canoncal d-connecton D s of Cartan type such that Cµc α = 0, then t s of Berwald type. Proof. Follows from the fact that 0 = P a µb = b N a µ Γ a bµ. Theorem 6.3. Let (T M, λ) be an EAP-space. Assume that the canoncal d-connecton D s of Berwald type. Then we have: (a) P a µb = 0 (b) λ µ are functons of the postonal argument x only. Consequently, so are g µν. (c) C α µc = 0. Consequently, γ α µc = 0. (d) The coeffcents Γ α µν and Γ α µν are gven respectvely by are functons of the postonal argument x only and Γ α µν(x) = ( λ α ν λ µ )(x), Γ α µν(x) = 1 2 gαǫ ( µ g νǫ + ν g µǫ ǫ g µν )(x). (e) Λ α µν and γα µν are functons of the postonal argument x only. (f) γ a bµ = P a bµ = 0. Consequently, Γ a bµ = Γ a bµ. Proof. The proof s straghtforward except for the relaton γbµ a = 0, whch can be proved n exactly the same manner as (f) of Theorem 5.3. Corollary 6.4. If the canoncal d-connecton D s of Berwald type, then D, D and D are also of Berwald type. In what follows, we assume that the canoncal d-connecton s of Berwald type. The next two results are mmedate consequences of the fact that P a µc = Cα µc = γα µc = γa bµ = 0, takng nto account Theorems 3.4, 3.5, 3.6, 4.2, 4.3 and 4.4. Proposton 6.5. The followng relatons hold: (a) R a bµν = γa bd Rd µν, P α βνc = S α βcd = 0, (b) R α βνµ = Λα µν β, Pα βµc = P α βµc = 0, Pa bµc = W a bµc, 22

23 (c) W a bνµ = W α βνc = W α βνc = Ŵα βνc = 0. Theorem 6.6. The ndependent non-vanshng W-tensors are gven by: (a) W α βνµ = (γ α βµ ν γα βν µ )+(γǫ βν γα ǫµ γ ǫ βµ γα ǫν) γ α βǫ Λǫ νµ, (b) W a bνc = γ a bc ν, (c) W α βνµ = Λα νµ β +Λǫ νµ Λα βǫ, (d) W a bdc = Ta dc b +Te dc Ta be. To sum up, the assumpton that the canoncal d-connecton beng of Berwald type mples that most of the purely horzontal geometrc objects of the EAP-space become functons of the postonal argument x only and concde wth the correspondng geometrc objects of the classcal AP-space. Moreover, the three other defned d-connectons turn out also to be of Berwald type. Fnally, n ths case, there are twelve non-vanshng curvature tensors and four ndependent W-tensors. We end ths secton wth the followng table (compare wth Table 3). Table 4: EAP-geometry under the Berwald type case Connecton Coeffcents Torson Curvature Canoncal Dual (Γ α µν, b Nν a, 0, Ca bc ) (Λα µν, Ra µν, 0, 0, Ta bc ) (0, 0, 0, 0, 0, 0) (Γ α νµ, b N a ν, 0, Ca cb ) ( Λα µν, Ra µν, 0, 0, Ta bc ) ( R α βµν, R a bµν, 0, Pα βµc, 0, S a bcd ) Symmetrc (Γ α (µν), b N a ν, 0, Ca (bc) ) (0, Ra µν, 0, 0, 0) ( R α βµν, R a bµν, 0, P α βµc, 0, Ŝa bcd ) Natural ( Γ α µν, b N a ν, 0, C a bc ) (0, R a µν, 0, 0, 0) ( R α βµν, R a bµν, 0, P a bµc, 0, S a bcd ) 7. Recoverng the classcal AP-space We now assume that the canoncal d-connecton D s both Cartan and Berwald type. In vew of Proposton 6.2, ths condton s equvalent to the (apparently weaker) condton that D s of Cartan type and Cµc α = 0. We show that n ths case the classcal AP-space emerges, n a natural way, as a specal case from the EAP-space. We refer to ths condton as the CB-condton. As easly checked, we have CB-condton: N a µ = yb Γ a bµ, yb C a bc = 0; Cα µc = 0. Theorem 7.1. Assume that the CB-condton holds. Then 23

24 (1) The four defned d-connectons of the EAP-space concde up to the hh-coeffcents. Moreover, these hh-coeffcents are functons of the postonal argument x only and are dentcal to the coeffcents of the correspondng four defned connectons n the classcal AP-space. (2) The torson of the canoncal d-connecton and the contorton of the EAP-space are functons of the postonal argument x only and are gven by T = (Λ α µν, 0, 0, 0, 0); C = (γ α µν, 0, 0, 0) (3) The three non-vanshng curvature tensors are functons of the postonal argument x only and are gven by (a) R α βµν = (γ α βµ ν γα βν µ )+(γǫ βν γα ǫµ γ ǫ βµ γα ǫν)+γ α βǫ Λǫ µν (b) R α βµν = Λα νµ β, (c) R α βµν = 1 2 (Λα βµ ν Λα βν µ )+ 1 4 (Λǫ βµ Λα νǫ Λ ǫ βν Λα µǫ)+ 1 2 (Λǫ µνλ α βǫ ) (4) There s only one W-tensor whch s a functon of the postonal argument x only and s gven by W α βνµ = Λα νµ β +Λǫ νµ Λα βǫ All other W-tensors vansh dentcally, or concde wth the correspondng curvature tensors. Consequently, the fundamental geometrc objects of the EAP-space concde wth the correspondng fundamental geometrc objects of the classcal AP-space [17]. Corollary 7.2. If the canoncal d-connecton D satsfes the CB-condton, then and D also satsfy the CB-condton. D, D We end ths secton wth the followng two tables whch summarze the geometry of the EAP-space under the CB-condton. Table 5: Fundamental connectons of EAP-space under the CB-condton Connecton Coeffcents hh-coeffcents Canoncal (Γ α µν, b Nν, a 0, Cbc a ) Γα µν(x) = ( λ α ν λ µ )(x) Dual ( Γ α µν, b N a ν, 0, Ca bc ) Γα µν (x) = Γ α νµ (x) Symmetrc ( Γ α µν, b N a ν, 0, Ca bc ) Γα µν (x) = Γ α (µν) (x) Natural ( Γ α µν, b N a ν, 0, C a bc ) Γ α µν (x) = 1 2 gαǫ ( µ g νǫ + ν g µǫ ǫ g µν )(x) 24

25 Table 6: EAP-geometry under the CB-condton Connecton Coeffcents Torson Curvature Canoncal Dual (Γ α µν, b N a ν, 0, C a bc ) (Λα µν, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0) (Γ α νµ, b Nν, a 0, Cbc a ) ( Λα µν, 0, 0, 0, 0) ( R βµν α, 0, 0, 0, 0, 0) Symmetrc (Γ α (µν), b Nν, a 0, Cbc a ) (0, 0, 0, 0, 0) ( R βµν α, 0, 0, 0, 0, 0) Natural ( Γ α µν, b N a ν, 0, C a bc ) (0, 0, 0, 0, 0) ( R α βµν, 0, 0, 0, 0, 0) Some comments on the CB-condton: (a) It should be noted that the non-vanshng of the purely vertcal tensors λ a, Cbc a, and g ab and the hv-coeffcents Γ a bµ of the canoncal d-connecton may represent extra degrees of freedom whch do not exst n the classcal AP-context. Moreover, these vertcal geometrc objects stll depend on the drectonal argument y. However, they actually do not contrbute to the EAP-geometry under the CB-condton. Ths s because the torson and the contorton tensors n ths case have only one non-vanshng counterpart each, namely the purely horzontal components Λ α µν and γµν α respectvely. (b) One readng of Theorem 7.1 s roughly that the projecton of the geometrc objects of the EAP-space on the base manfold M can be dentfed wth the classcal AP-geometry. The dstncton whch appears between the two geometres s due to the fact that the geometrc objects of the EAP-space lve n the double tangent bundle TTM TM and not n the tangent bundle TM M. Consequently, t can be sad, roughly speakng, that the classcal AP-space s a copy of the EAPspace equpped wth the CB-condton, vewed from the perspectve of the base manfold M. 8. Concludng remarks In the present artcle, we have constructed and developed a parallelzable structure analogous to the AP-geometry on the tangent bundle T M of M. Four lnear connectons, dependng on one a pror gven nonlnear connecton, are used to explore the propertes of ths geometry. Dfferent curvature tensors charaterzng ths structure, together wth ther contractons, are computed. The dfferent W-tensors are also derved. Extra condtons are mposed on the canoncal d-connecton, the consequences of whch are nvestgated. Fnally, a specal form of the canoncal d-connecton s ntroduced under whch the EAP-geometry reduces to the classcal AP-geometry. On the present work, we have the followng comments: 25

26 (1) Exstng theores of gravty suffer from some problems connected to recent observed astrophyscal phenomena, especally those admttng ansotropc behavor of the system concerned (e.g. the flatness of the rotaton curves of spral galaxes). So, theores n whch the gravtatonal potental depends on both poston and drecton may be needed. Such theores are to be constructed n spaces admttng ths dependence; a potental canddate s the EAP-space. Ths s one of the ams motvatng the present work. (2) One possble physcal applcaton of the EAP-geometry would be the constructon of a generalzed feld theory on the tangent bundle TM of M. Ths could be acheved by a double contracton of the purely horzontal W-tensor W αβ µν and the purely vertcal W-tensor W ab cd to obtan respectvely the horzontal scalar Lagrangan H := λ H := λ g αβ H αβ, where λ := det(λ α ) and H αβ := Λ ν ǫα Λǫ νβ C αc β and the vertcal scalar Lagrangan V := λ V := λ g ab V ab, where λ := det(λ a ) and V ab := T d eat e db C a C b. The feld equatons are obtaned by the use of the Euler-Lagrange equatons H H λ β xǫ( ) H λ β,ǫ ye( ) = 0, λ β;e (horzontal form) V V λ b xǫ( ) V λ b,ǫ ye( ) = 0. (vertcal form) λ b;e The resultng feld equatons could be compared wth those derved by M. Wanas [12] and R. Mron [6]. (3) Among the advantages of the classcal AP-geometry are the ease n calculatons and the dverse and ts thorough applcatons [14]. For these reasons, we have kept, n ths work, as close as possble to the classcal AP-case. However, the extra degrees of freedom n our EAP-geometry have created an abundance of geometrc objects whch have no counterpart n the classcal AP-geometry. Snce the physcal meanng of most of the geometrc objects of the classcal AP-structure s clear, we hope to attrbute physcal meanng to the new geometrc objects appearng n the present work. The physcal nterpretaton of the geometrc objects exstng n the EAP-space and not n the AP-geometry may need deeper nvestgaton. The study of the Cartan type case, due to ts smplcty, may be our frst step n tacklng the general case. (4) In concluson, we hope that physcts workng n AP-geometry would dvert ther attenton to the study of EAP-geometry and ts consequences, due to ts wealth, relatve smplcty and ts close resemblance (at least n form) to the classcal AP-geometry. We beleve that the extra degrees of freedom offered by the EAPgeometry may gve us more nsght nto the nfrastructure of physcal phenomena studed n the context of classcal AP-geometry and thus help us better understand the theory of general relatvty and ts connecton to other physcal theores. 26