On Curvature Tensors in Absolute Parallelism Geometry

Transcript

1 arxv:gr-qc/ v1 26 Apr 2006 On Curvature Tensors n Absolute Parallelsm Geometry Nabl L. Youssef and Amr M. Sd-Ahmed Department of Mathematcs, Faculty of Scence, Caro Unversty e-mal: nyoussef@frcu.eun.eg e-mal: amrs@maler.eun.eg Abstract In ths paper we dscuss curvature tensors n the context of Absolute Parallelsm geometry. Dfferent curvature tensors are expressed n a compact form n terms of the torson tensor of the canoncal connecton. Usng the Banch denttes some other denttes are derved from the expressons obtaned. These denttes, n turn, are used to reveal some of the propertes satsfed by an ntrgung fourth order tensor whch we refer to as Wanas tensor. A further condton on the canoncal connecton s mposed, assumng t s sem-symmetrc. The formulae thus obtaned, together wth other formulae (Rcc tensors and scalar curvatures of the dfferent connectons admtted by the space) are calculated under ths addtonal assumpton. Consderng a specfc form of the sem-symmetrc connecton causes all nonvanshng curvature tensors to concde, up to a constant, wth the Wanas tensor. Physcal aspects of some of the geometrc objects consdered are mentoned. 1 Introducton After the success of hs general theory of gravtaton (GR), Ensten searched for a more general theory that would approprately descrbe electromagnetc phenomena together wth gravty. Hs search for such a unfed theory led hm to consder Absolute Parallelsm (AP) geometry [3]. The reason for ths s that AP-geometry s wder than the standard Remannan geometry. Accordng to GR, the ten degrees of freedom (the metrc components for n = 4) are just suffcent to descrbe gravtatonal phenomena alone. AP-geometry, on the other hand, has sxteen degrees of freedom. Remannan geometry s thus relatvely 1

2 lmted compared to AP-geometry whch has sx extra degrees of freedom. These extra degrees of freedom could be used to descrbe physcal phenomena other than gravty. Ths dea of ncreasng the number of degrees of freedom from ten to sxteen s another alternatve to the dea of ncreasng the dmensons of the underlyng manfold n the so-called Kaluza-Klen theory. Moreover, as opposed to Remannan geometry whch admts only one symmetrc affne connecton and hence only one curvature tensor, the AP-space admts at least four bult-n (natural) affne connectons, two of whch are non-symmetrc. AP-geometry also admts tensors of thrd order, a number of second order skew and symmetrc tensors and a non-vanshng torson [8]. These extra geometrc structures, whch have no counterpart n Remannan geometry, make AP-geometry much rcher n ts content and hence a potental canddate for geometrc unfcaton schemes. A further advantage s that assocated to an AP-space there s a Remannan structure defned n a natural way; thus the AP-space contans wthn ts geometrc structure all the mathematcal machnery of Remannan geometry. Ths facltates comparson between the results obtaned n the context of AP-geometry wth the classcal GR based on Remannan geometry. Fnally, calculatons wthn the framework of AP-geometry are relatvely easer than those used n the context of Remannan geometry. In the present paper we nvestgate the curvature tensors correspondng to the dfferent natural connectons defned n an AP-space. The paper s organzed n the followng manner. In secton 2, we gve a bref account of the basc elements of AP-geometry; we focus our attenton on the fundamental concepts that wll be needed n the sequel. In secton 3, we consder the curvature tensors of the dual and symmetrc connectons assocated to the canoncal connecton Γ together wth the Remannan curvature. Smple and compact expressons for such curvature tensors, n terms of the torson of Γ, are deduced. We then use the Banch denttes to derve some further nterestng denttes whch smplfy the formulae thus obtaned. In secton 4, we study some of the propertes satsfed by an ntrgung fourth order tensor, whch we call Wanas tensor (or smply W-tensor) after M.I. Wanas who frst defned and used such a tensor [11]. The W-tensor s shown to have some propertes whch are smlar to those of the Remannan curvature tensor. In secton 5, we further study the consequences of assumng that the canoncal connecton s sem-symmetrc. All curvature tensors wth ther assocated contractons (Rcc tensors and scalar curvatures) are then derved n ths case. We next consder an nterestng specal case whch consderably smplfes the formulae thus obtaned and show that all nonvanshng curvature tensors admtted by the space concde (up to a constant factor) wth the W-tensor. Physcal aspects or physcal nterpretatons of some geometrc objects consdered are ponted out. 2

3 2 A Bref account of AP geometry In ths secton, we gve a bref account of the geometry of absolute parallelsm spaces. For more detals concernng ths geometry, we refer for example to [8], [13], [10]. An absolute parallelsm space (AP-space) s an n-dmensonal C manfold M whch admts n lnearly ndependent global vector felds λ ( = 1,...,n) on M. Such a space s also known n the lterature as parallelzable manfold [2]. Let λ µ (µ = 1,...,n) be the coordnate components of the -th vector feldλ. The summaton conventon s appled on both Latn (mesh) and Greek (world) ndces. The covarant components of λ µ are gven va the relatons The n 3 functons Γ α µν defned by λ µ λ ν = δ ν µ, λ µ λ µ = δ j. j Γ α µν := λ α λ µ,ν transform as the coeffcents of a lnear connecton under a change of coordnates (where, ν denotes partal dfferentaton wth respect to the coordnate x ν ). The connecton Γ α µν s clearly non-symmetrc and s referred to as the canoncal connecton of the space. As easly checked, we have λ µ ν = 0, λ µ ν = 0, where the stroke denotes covarant dfferentaton wth respect to the canoncal connecton Γ α µν. The above relaton s known n the lterature as the condton of absolute parallelsm. Let Λ α µν := Γα µν Γα νµ denote the torson tensor of Γα µν. It s of partcular mportance to note that the condton of absolute parallelsm together wth the commutaton formula λ α µν λ α νµ = λ ǫ Rǫµν α + λ α ǫ Λ ǫ νµ ; forces the curvature tensor R α µνσ of the canoncal connecton Γ α µν to vansh dentcally. It s for ths reason that many authors thnk that the AP-space s a flat space. Ths s by no means true. In fact, t s meanngless to speak of curvature wthout reference to a connecton. All we can say s that the AP-space s flat wth respect to ts canoncal connecton, or that ts canoncal connecton s flat. However, there are other three natural (bult-n) connectons whch are nonflat. Namely, the dual connecton Γ α µν := Γα νµ, 3

4 the symmetrc connecton ˆΓ α µν := 1 2 (Γα µν + Γα νµ ) = Γα (µν) and the Remannan connecton (Chrstoffel symbol) Γ α µν := 1 2 {gαǫ (g ǫν,µ + g ǫµ,ν g µν,ǫ )} assocated to the metrc structure defned by wth nverse g µν := λ µ λ ν g µν = λ µ It s to be noted that the condton of absolute parallelsm mples that the canoncal connecton Γ α µν s metrc: λ ν. g µν σ = 0, g µν σ = 0. Consequently, the covarant dfferentaton defned by the canoncal connecton commutes wth contracton by the metrc tensor g µν and ts nverse g µν. The contorton tensor s defned by γ α µν := Γα µν Γ α µν. Snce Γ α µν s symmetrc, t follows that Moreover, t can be shown that [8] Λ α µν = γα µν γα νµ. γ α µν = λ α λ µ o ν, where denotes covarant dfferentaton wth respect to the connecton Γ α µν. Fnally, the contorton tensor can be expressed n terms of the torson tensor n the form [4]: γ µνσ = 1 2 (Λ µνσ + Λ σνµ + Λ νσµ ), where γ µνσ = g ǫµ γ ǫ νσ and Λ µνσ = g ǫµ Λ ǫ νσ. It s to be noted that Λ µνσ s skew-symmetrc n the last par of ndces, whereas γ µνσ s skew symmetrc n the frst par of ndces. It s to be noted also that the contorton tensor vanshes f and only f the torson tensor vanshes. 4

5 We have four types of covarant dervatves correspondng to the four connectons mentoned above, namely A µ ν = A µ,ν + Γ µ ǫν Aǫ, A µ ν = A µ,ν + Γ µ ǫν Aǫ, A µ ˆ ν = A µ,ν + ˆΓ µ ǫνa ǫ, A µ o ν = A µ,ν + Γ µ ǫνa ǫ, where A µ s an arbtrary contravarant vector. In concluson, the AP-space has four curvature tensors R α µνσ, R α µνσ, ˆR α µνσ and R α µνσ correspondng to the four connectons Γ α µν, Γ α µν, ˆΓ α µν and Γ α µν respectvely. As already mentoned, only one of these curvature tensors vanshes dentcally (the curvature Rµνσ α of the canoncal connecton). The other three do not vansh n general. The vanshng of Rµνσ α enables us to express the other three curvature tensors n terms of the torson tensor Λ α µν as wll be revealed n the next secton. Summary of the geometry of the AP-space Connecton Coeffcents Covarant Torson Curvature Metrcty dervatve Canoncal Γ α µν Λ α µ ν 0 metrc Dual Γα µν Λ α µ ν R α µνσ non-metrc Symmetrc ˆΓα µν ˆ 0 ˆRα µνσ non-metrc Remannan Γ α µν 0 R α µνσ metrc 5

6 3 Curvature Tensors and the Banch denttes n AP-geometry Let (M, λ) be an AP-space of dmenson n where λ ( = 1,...,n) are the n lnearly ndependent vector felds defnng the AP-structure on M. Let Γ α µν be the canoncal connecton on M defned by Γ α µν = λ α λ µ,ν. Let Γ α µν, ˆΓ α µν and Γ α µν be respectvely the dual connecton assocated to Γ α µν, the symmetrc connecton assocated to Γα µν and the Remannan connecton defned by the metrc tensor g µν = λ µ λ ν. As n the prevous secton, covarant dfferentaton wth respect to Γ α µν, Γ α µν ˆΓ α µν and Γ α µν wll be denoted by,, ˆ and respectvely. Theorem 3.1. The curvature tensors R α µνσ, ˆRα µνσ and R α µνσ of the connectons Γ α µν, ˆΓ α µν and Γ α µν are expressed n terms of the torson tensor Λ α µν of the canoncal connecton Γ α µν as follows: (a) R α µνσ = Λ α σν µ. (b) ˆR α µνσ = 1 2 (Λα µν σ Λα µσ ν ) (Λǫ µνλ α σǫ Λ ǫ µσλ α νǫ) (Λǫ σνλ α ǫµ). (c) R α µνσ = (γα µν σ γα µσ ν ) + (γǫ µσ γα ǫν γǫ µν γα ǫσ ) + γα µǫ Λǫ νσ. Proof. We start by provng the frst relaton: R α µνσ = Γ α µσ,ν Γ α µν,σ + Γ ǫ µσ Γ α ǫν Γ ǫ µν Γ α ǫσ = Γ α σµ,ν Γ α νµ,σ + Γ ǫ σµγ α νǫ Γ ǫ νµγ α σǫ = {Γ α σµ,ν + Γǫ σµ (Λα νǫ + Γα νǫ )} {Γα νµ,σ + Γǫ νµ (Λα σǫ + Γα ǫσ )} = (Γ α σµ,ν + Γǫ σµ Γα ǫν ) (Γα νµ,σ + Γǫ νµ Γα ǫσ ) (Γǫ σµ Λα ǫν + Γǫ νµ Λα σǫ ) = (R α σνµ + Γ α σν,µ + Γ ǫ σνγ α ǫµ) + (R α νµσ Γ α νσ,µ Γ ǫ νσγ α ǫµ) (Γ ǫ σµλ α ǫν + Γ ǫ νµλ α σǫ). Takng nto account the fact that Rµνσ α = 0, we get R µνσ α = Λα σν,µ + Γα ǫµ Λǫ σν Γǫ σµ Λα ǫν Γǫ νµ Λα σǫ = Λα σν µ. We then prove the second relaton: We have, by defnton, ˆR µνσ α = ˆΓ α µσ,ν ˆΓ α µν,σ + ˆΓ ǫ µσˆγ α ǫν ˆΓ ǫ µνˆγ α ǫσ. Now, ˆΓ ǫ µσˆγ α ǫν = 1 4 (Λǫ σµ + 2Γǫ µσ )(Λα νǫ + 2Γα ǫν ) = 1 4 Λǫ µσλ α νǫ 1 2 Λǫ µσγ α ǫν 1 2 Λα ǫνγ ǫ µσ + Γ ǫ µσγ α ǫν. 6

7 Smlarly, ˆΓ ǫ µνˆγ α ǫσ = 1 4 Λǫ µν Λα σǫ 1 2 Λǫ µν Γα ǫσ 1 2 Λα ǫσ Γǫ µν + Γǫ µν Γα ǫσ. Moreover, ˆΓ α µσ,ν = 1 2 Λα µσ,ν + Γα µσ,ν and ˆΓ α µν,σ = 1 2 Λα µν,σ + Γα µν,σ. Hence, notng that Rµνσ α = 0, we get ˆR α µνσ = 1 4 (Λǫ µνλ α σǫ Λ ǫ µσλ α νǫ) {(Λα µν,σ + Λ ǫ µνγ α ǫσ Λ α ǫνγ ǫ µσ) (Λ α µσ,ν + Λǫ µσ Γα ǫν Λα ǫσ Γǫ µν )} = 1 2 (Λα µν σ Λα µσ ν ) (Λǫ µν Λα σǫ Λǫ µσ Λα νǫ ) (Λǫ σν Λα ǫµ ). The proof of relaton (c) s carred out n the same manner and we omt t. It s clear that the torson tensor plays the key role n all denttes obtaned above. The vanshng of the torson tensor forces the three connectons Γ α µν, Γ α µν and ˆΓ α µν to concde wth the Remannan connecton Γ α µν ; the Remannan space n ths case s trvally flat. Consequently, the non-vanshng of any of the three curvature tensors suffces for the non-vanshng of the torson tensor. Remark 3.2. The frst and second formulae (resp. The thrd formula) of the above theorem reman (resp. remans) vald n the more general context n whch Γ α µν s any gven non-symmetrc lnear connecton on a manfold M (resp. a Remannan manfold (M, g)) wth vanshng curvature. We now derve some relatons that wll prove useful later on. Proposton 3.3. The followng relatons hold: (a) Λ α µν σ Λα µν σ = S µ,ν,σ Λ ǫ µν Λα ǫσ. (b) Λ α µν σ Λα µνˆ σ = 1 2 (S µ,ν,σ Λ ǫ µν Λα ǫσ ). (c) Λ α µν σ Λα µν o σ = Λǫ µν γα ǫσ + Λα νǫ γǫ µσ + Λα ǫµ γǫ νσ. where the notaton S µ,ν,σ denotes a cyclc permutaton of the ndces µ, ν, σ and summaton. Proof. The three relatons follow from the defnton of the covarant dervatve wth respect to the approprate connecton. Let M be a dfferentable manfold equpped wth a lnear connecton wth torson T and curvature R. Then the Banch denttes are gven locally by [7] 7

8 (a) S µ,ν,σ R α µνσ = S µ,ν,σ (T α µν;σ + T ǫ µνt α ǫσ), (frst Banch dentty) (b) S µ,ν,σ (R α βµν;σ + Rα βµǫ T ǫ νσ) = 0. (second Banch dentty) where ; denotes covarant dfferentaton wth respect to the gven connecton. In what follows, we derve some denttes usng the frst and second Banch denttes. Some of the derved denttes wll be used n smplfyng some of the formulae thus obtaned. Theorem 3.4. The frst Banch dentty for the connectons Γ α µν, Γ α µν, ˆΓ α µν and Γ α µν reads: (a) S µ,ν,σ (Λ α µν σ + Λǫ µνλ α ǫσ) = 0. (b) S µ,ν,σ Rα µνσ = S µ,ν,σ (Λ α νµ σ + Λǫ µν Λα ǫσ ). (c) S µ,ν,σ ˆRα µνσ = 0. (d) S µ,ν,σ R α µνσ = 0. The second Banch dentty for the connecton Γ α µν, ˆΓ α µν and Γ α µν reads: (e) S µ,ν,σ Rα βµν σ = S µ,ν,σ Λ ǫ σνλ α ǫµ β. (f) S µ,ν,σ ˆRα βµνˆ σ = 0. (g) S µ,ν,σ R α βµν o σ = 0. Proof. Identtes (a) and (b) follow respectvely from the fact that R α µνσ vanshes dentcally and that Λ α µν = Λ α νµ. Identty (e) results from Theorem 3.1 (a) together wth the fact that Λ α µν = Λ α νµ. The remanng denttes are trval because of the symmetry of the connectons ˆΓ α µν and Γ α µν. Proposton 3.5. The followng denttes hold: (a) S µ,ν,σ Λ α µν σ = 0. (b) S µ,ν,σ Rα µνσ = S µ,ν,σ (Λ ǫ µνλ α ǫσ). Proof. Takng nto account Theorem 3.4 (b) together wth Theorem 3.1 (a) we get S µ,ν,σ (Λ α νσ µ + Λǫ µν Λα ǫσ ) = S µ,ν,σ Λ α µν σ. By Theorem 3.4 (a), the left hand sde of the the above equaton vanshes and the frst dentty follows. The second dentty s a drect consequence of theorem 3.4 (b), takng nto consderaton dentty (a) above. The next result s crucal n smplfyng some of the denttes obtaned so far and n provng other nterestng results. 8

9 Theorem 3.6. The torson tensor satsfes the dentty S µ,ν,σ Λ ǫ µν Λα ǫσ = 0. Proof. Applyng the frst Banch dentty to ˆR µνσ α Theorem 3.1 (b), we get as expressed n 1 2 S µ,ν,σ (Λ α µν σ Λα µσ ν )+1 4 S µ,ν,σ (Λ ǫ µνλ α σǫ Λ ǫ µσλ α νǫ)+ 1 2 S µ,ν,σ Λ ǫ νσλ α µǫ = 0. ( ) Consderng each of the above three terms separately, and takng nto account Theorem 3.4 (a), we obtan respectvely 1 2 S µ,ν,σ (Λ α µν σ Λα µσ ν ) = 1 2 S µ,ν,σ (Λ α µν σ +Λα σµ ν ) = S µ,ν,σ Λ α µν σ = S µ,ν,σ Λ ǫ µν Λα σǫ, 1 4 S µ,ν,σ (Λ ǫ µν Λα σǫ Λǫ µσ Λα νǫ ) = 1 4 S µ,ν,σ (Λ ǫ µν Λα σǫ +Λǫ σµ Λα νǫ ) = 1 2 S µ,ν,σ Λ ǫ µν Λα σǫ and 1 2 S µ,ν,σ Λ ǫ νσ Λα µǫ = 1 2 S µ,ν,σ Λ ǫ µν Λα σǫ. The requred dentty results by substtutng the above three equatons nto ( ). Corollary 3.7. The followng denttes hold: (a) Λ α µν σ = Λα µν σ = Λα µνˆ σ. (b) S µ,ν,σ Λ α µν σ = 0. (c) S µ,ν,σ Rα µνσ = 0. (d) S µ,ν,σ, Λ α νµ βσ = S µ,ν,σ Λ ǫ σν Λα ǫµ β (Λα νµ βσ := Λα νµ β σ ). (e) S µ,ν,σ (γ α µν σ γα νµ σ ) = 0. Proof. Takng nto account Theorem 3.6, relaton (a) follows from Proposton 3.3 (a) and (b), whereas denttes (b) and (c) follow from Theorem 3.4 (a) and Proposton 3.5 (b) resp. Identty (d) follows from Theorem 3.4 (e) takng nto account Theorem 3.1 (a) together wth relaton (a) above. Fnally, the last dentty follows from dentty (b) above together wth the relaton Λ α µν = γα µν γα νµ. In vew of Theorem 3.6, the curvature tensor ˆR α µνσ can be further smplfed as revealed n Proposton 3.8. The curvature tensor ˆR µνσ α can be expressed n the form: ˆR µνσ α = 1 2 Λα σν µ Λǫ σν Λα ǫµ. 9

10 Proof. The curvature tensor ˆR α µνσ has the form (Theorem 3.1 (b)): ˆR α µνσ = 1 2 (Λα µν σ Λα µσ ν ) (Λǫ µν Λα σǫ Λǫ µσ Λα νǫ ) (Λǫ σν Λα ǫµ ). Takng nto account dentty (b) n Corollary 3.7 and Theorem 3.6, we get ˆR α µνσ = 1 2 {(S µ,ν,σ Λ α µν σ ) Λα νσ µ } Λǫ νσλ α µǫ {(S µ,ν,ǫ Λ ǫ µν Λα σǫ ) Λǫ νσ Λα µǫ } = 1 2 Λα σν µ Λǫ σνλ α ǫµ. The formula obtaned for the curvature tensors R α µνσ n Theorem 3.1 s strkngly compact and elegant. The formula obtaned for ˆR α µνσ s however less elegant but stll relatvely compact. These two formulae, together wth the frst Banch dentty, enabled us to derve, n a smple way, the crucal denttes S µ,ν,σ Λ α µν σ = 0 and S µ,ν,σ Λ ǫ µν Λα ǫσ = 0 whch, n turn, smplfed the formula obtaned for ˆR α µνσ (whch s now more elegant). The last two denttes wll play an essental role n the rest of the paper. It should be noted however that a drect proof of these denttes s far from clear. Remark 3.9. All results and denttes concernng the connectons Γ α µν, Γ α µν and ˆΓ α µν (resp. the connecton Γ α µν) reman vald n the more general context n whch Γ α µν s a non-symmetrc lnear connecton on a manfold M (resp. a Remannan manfold (M, g)) wth vanshng curvature. 10

11 4 The Wanas Tensor (W-tensor) Let (M, λ) be an AP-space of dmenson n, whereλ are the n lnearly ndependent vector felds defnng the AP-structure on the manfold M. Let R µνσ α and Λ α µν be the curvature and the torson tensors of the dual connecton Γ α µν. Defnton 4.1. The tensor feld Wµνσ α of type (1,3) on M defned by the formula λ ǫ Wµνσ ǫ := λ µ νσ λ µ σν wll be called the Wanas tensor, or smply the W-tensor, of the AP-space (M, λ). The W-tensor has been frst defned by M.I Wanas [11] n 1975 and has been used by F. Mkhal and M. Wanas [9] n 1977 to construct a geometrc theory unfyng gravty and electromagnetsm. The next result gves qute a smple expresson for such a tensor. Theorem 4.2. Let (M, λ) be an AP-space. Then the W-tensor can be expressed n the form W α µνσ = Λα σν µ Λǫ σν Λα ǫµ, where Λ α µν s the torson tensor of the canoncal connecton Γα µν. Proof. Consder the commutaton formula wth respect to the connecton Γ α µν: λ µ νσ λ µ σν = λ ǫ Rǫ µνσ + λ µ ǫ Λǫ νσ. Multplyng both sdes by λ α, usng the defnton of the W-tensor together wth the defnton of Γ α µν and takng nto account Theorem 3.1 (a), we get W α µνσ = Λ α σν µ + λ α ( λ µ,ǫ λ β Γ β ǫµ)λ ǫ σν = Λ α σν µ + (Γα µǫ Γα ǫµ )Λǫ σν = Λ α σν µ Λǫ σνλ α ǫµ. Remark 4.3. The W-tensor can be also defned contravarantly as follows: λ µ Wµνσ α = λ α λ νσ α = λ σν ǫ Rα ǫσν + λ α Λǫ ǫ νσ. Ths defnton gves the same formula for the W-tensor as n Theorem 4.2. Proposton 4.4. The Wanas tensor has the followng propertes: (a) W α µνσ s skew symmetrc n the last par of ndces. 11

12 (b) W α µνσ β W α µνσ β = Λα σν µβ Λα σν µβ. Proof. The frst property s trval. The second property holds as a result of Theorem 4.2 together wth corollary 3.7 (a). Theorem 4.5. The W-tensor satsfes the followng dentty: S µ,ν,σ W α µνσ = 0. Proof. Follows drectly from Theorem 4.2, takng nto account Theorem 3.6 together wth Corollary 3.7 (b). The dentty satsfed by the W-tensor n Theorem 4.5 s the same as frst Banch dentty of the Remannan curvature tensor. The dentty correspondng to the second Banch dentty s gven by: Theorem 4.6. The W-tensor satsfes the followng dentty: S ν,σ,β W α µνσ β = S ν,σ,β {Λ ǫ νσ(λ α ǫβ µ + Λα ǫµ β )}. Proof. Takng nto account the second Banch dentty, Theorem 4.2, Theorem 3.1 (a) and Corollary 3.8 (a) and (b), we get S ν,σ,β W α µνσ β = S ν,σ,β { R α µνσ β (Λǫ σν Λα ǫµ ) β } = S ν,σ,β ( R α µνǫ Λ ǫ βσ Λǫ σν Λα ǫµ β ) = S ν,σ,β (Λ α ǫν µ Λǫ σβ Λα ǫµ β Λǫ σν ) = S ν,σ,β {Λ ǫ νσ(λ α ǫβ µ + Λα ǫµ β )}. We conclude ths secton by some comments: (1) The W-tensor s unque n the sense that t can be defned only n terms of the connecton Γ α µν. The same defnton usng the three other connectons gves nothng new. In fact, the commutaton formula for the connecton Γ α νµ s trval (snce λ α µ = 0), whereas the commutaton formulae for ˆΓ α µν and Γ α µν gve rse to ˆR α νµσ and R α µνσ respectvely (snce the torson tensor vanshes n the latter two cases.) (2) The vanshng of the torson tensor mples the vanshng of the W-tensor whch s equvalent to the commutatvty of successve covarant dfferentaton (of the vector feldsλ); a strkng property whch does not exst n Remannan geometry (or even n other geometres, n general). (3) The W-tensor has some propertes common wth the Remannan curvature tensor (for example, Proposton 4.4 (a) and Theorem 4.5). Nevertheless, there are sgnfcant devatons from the Remannan curvature tensor (for example, the vanshng of the W- tensor does not necessarly mply flatness). 12

13 (4) The expresson of the W-tensor comprses, n addton to the term contanng the curvature tensor R α µνσ, a term contanng a torson contrbuton. Thus, roughly speakng, the W-tensor expresses geometrcally the nteracton between curvature and torson. On the other hand, as gravty s descrbed n terms of curvature and electromagnetsm s descrbed n terms of torson [12], we can roughly say that the W-tensor expresses physcally the nteracton between gravty and electromagnetsm [9]. 5 The Sem-Symmetrc Case Defnton 5.1. Let M be a dfferentable manfold. A sem-symmetrc connecton on M s a lnear connecton on M whose torson tensor T s gven by [14] T α µν = δ α µw ν δ α ν w µ, for some scalar 1-form w µ. Sem-symmetrc connectons have been studed by many authors (cf. for example [1], [5], [15], [16]). In what follows, we consder an n- dmensonal AP-space (M, λ) wth the addtonal assumpton that the canoncal connecton Γ α µν s sem-symmetrc. Then, by Defnton 5.1, we have Λ α µν = δµw α ν δν α w µ. Moreover, t can be shown that [6] Γ α µν = Γ α µν + δ α µw ν g µν w α, where Γ α µν s the Remannan connecton defned n secton 2 and w α = g αµ w µ. Hence, γ α µν = δ α µw ν g µν w α. Theorem 5.2. The curvature tensors R α µνσ, ˆRα µνσ and R α µνσ of the connectons Γ α µν, ˆΓ α µν and Γ α µν are expressed n terms of w µ n the form (a) R α µνσ = δα σ w ν µ δ α ν w σ µ. (b) ˆR α µνσ = 1 2 (δα σ w ν µ δ α ν w σ µ) w µ(δ α σ w ν δ α ν w σ). (c) R α µνσ = δα µ (w ν σ w σ ν ) + (g µσ w α ν g µν w α σ) + 2w α (g µσ w ν g µν w σ ). Consequently, f R αµνσ = g αǫ Rǫ µνσ wth smlar expressons for ˆR αµνσ and R αµνσ, we get (a) Rαµνσ = g ασ w ν µ g αν w σ µ. 13

14 (b) ˆRαµνσ = 1 2 (g ασw ν µ g αν w σ µ ) w µ(g ασ w ν g αν w σ ). (c) Rαµνσ = g αµ (w ν σ w σ ν )+(g µσ w α ν g µν w α σ )+2w α (g µσ w ν g µν w σ ). Proof. The frst two relatons hold by substtutng Λ α µν n the formulae expressng R µνσ α (Theorem 3.1 (a)) and ˆR µνσ α (Proposton 3.8) respectvely. The thrd relaton holds by substtutng Λ α µν and γµν α n the formula expressng R α µνσ (Theorem 3.1 (c)). The remanng relatons are straghtforward. Proposton 5.3. Let Rµν = R α µνα and R = g µν Rµν, wth smlar expressons for ˆRα µνσ and R α µνα. Then (a) R µν = (n 1)w ν µ, R = (n 1)w µ µ. (b) ˆR µν = 1 4 (n 1)(2w ν µ + w µ w ν ), ˆR = 1 4 (n 1)(2wµ µ + w µ w µ ). (c) R µν = w ν µ g µν w σ σ + 2(w ν w µ g µν w σ w σ ), 2w µ w µ ). R = (1 n)(w µ µ + Proof. Follows drectly from the relatons obtaned n Theorem 5.2 by applyng the sutable contractons. Theorem 5.4. The second order covarant tensor w ν µ s symmetrc: w µ ν = w ν µ. Proof. Follows drectly from Proposton 5.3 (c) notng that R µν s symmetrc, beng the Rcc tensor of the Remannan connecton. Ths can be also deduced from Theorem 5.2 (c) notng that R µ µνσ = 0. A drect consequence of the above theorem s the followng Corollary 5.5. The followng propertes hold: (a) R µν and ˆR µν are symmetrc. (b) R µ µνσ = 0. (c) ˆR µ µνσ = 0. Theorem 5.6. The W-tensor has the form (a) W α µνσ = δα σ (w ν µ w µ w ν ) δ α ν (w σ µ w σ w µ ). (b) W αµνσ = g ασ (w ν µ w µ w ν ) g αν (w σ µ w σ w µ ) (W αµνσ = g αǫ W ǫ µνσ ). Proof. Follows drectly from Theorem 4.2 by substtutng the expresson of Λ α µν n terms w µ. Corollary 5.7. Let W µν = W ǫ µνǫ and W = g µν W µν. Then 14

15 (a) W µν = (n 1)(w ν µ w µ w ν ). (b) W = (n 1)(w µ µ w µ w µ ). (c) W µ µνσ = 0. Consequently, W µν s symmetrc. Comparng Corollary 5.7 (a) and Theorem 5.6 (a), we obtan Corollary 5.8. Let dm M 2. A suffcent condton for the vanshng of the W-tensor s that W µν = 0. Fnally, n vew of Theorem 1 of K. Yano [14], the followng result follows: Theorem 5.9. If the canoncal connecton Γ α µν of an AP-space s semsymmetrc, then the assocated Remannan metrc g µν s conformally flat. Specal case. In the followng we assume that the canoncal connecton Γ α µν s a sem-symmetrc connecton whose defnng 1-form w µ satsfes the condton w µ w ν = δ ν µ. It s easy to see that the above condton mples that w µ ν = 0 and that w µ w ν = g µν. Under the gven assumpton, the dfferent curvature tensors and the Wanas tensor, accordng to Theorems 5.2, 5.6, take the form R α µνσ = 0, ˆR α µνσ = 1 4 (δα σ g µν δ α ν g µσ ), R α µνσ = 2(δα ν g µσ δ α σ g µν), W α µνσ = δα ν g µσ δ α σ g µν. Consequently, W α µνσ = 4 ˆR α µνσ = 1 2 R α µνσ and Rαµνσ = 2(g αν g µσ g ασ g µν ). In ths case, the W-tensor becomes a curvature-lke tensor [7] and the above formulae mply the followng result: 15

16 Theorem Let the canoncal connecton Γ α µν of an AP-space (M, λ) be a sem-symmetrc connecton whose defnng 1-form w µ satsfes the condton w µ w ν = δµ. ν Then, all nonvanshng curvature tensors of (M, λ) concde, up to a constant, wth the W-tensor and the AP-space becomes a Remannan space of constant curvature. It should be noted that, n ths partcular case, the AP-character of the space recedes, whereas the Remannan aspects of the AP-space become domnant. Physcally speakng, the latter result seems to suggest that, n ths partcular case, electromagnetc effects are absent. Ths partcular case can thus be consdered, n some sense, as a lmtng case. References [1] O. C. Andone et D. Smaranda. Certanes connexons semsymétrque. Tensor, N.S., Vol. 31 (1977) [2] F. Brckell and R. S. Clark. Dfferentable manfolds. Van Nostrand Renhold Co., [3] A. Ensten. Unfed feld theory based on Remannan metrcs and dstant parallelsm. Math. Annal., 102 (1930), [4] K. Hayash and T. Shrafuj. New general relatvty. Phys. Rev. D 19 (1979), [5] T. Ima. Notes on sem-symmetrc metrc connectons. Tensor, N.S., Vol. 24 (1972), [6] T. Ima. Notes on sem-symmetrc metrc connectons. Tensor, N.S., Vol. 27 (1973), [7] S. Kobayash and K. Nomzu. Foundatons of dfferental geometry. Vol. I. Interscence Publshers, [8] F. I. Mkhal. Tedrad vector felds and generalzng the theory of relatvty. An Shams Sc. Bull., No. 6 (1962), [9] F. I. Mkhal and M.I. Wanas. A generalzed feld theory, I. Feld equatons. Proc. R. Soc. London, A. 356 (1977), [10] H. P. Robertson. Groups of moton n spaces admttng absolute parallelsm. Ann. Math, Prnceton (2), 33 (1932), [11] M. I. Wanas. A generalzed feld theory and ts applcatons n cosmology. Ph.D Thess. Caro Unversty, [12] M. I. Wanas and S.A. Ammar. Space-tme structure and electromagnetsm. Electroncally avalable at arxv.gr-qc/

17 [13] M. I. Wanas. Absolute parallelsm geometry: Developments, applcatons and problems. Stud. Cercet, Stn. Ser. Mat. Unv. Bacau, No. 10, (2001) [14] K. Yano. On sem-symmetrc metrc connecton. Rev. Roumane Math. Pures. Appl., T. 15 (1970), [15] N. L. Youssef. Connexons métrque sem-symétrques sembasques. Tensor, N.S., Vol. 40 (1983), [16] N. L. Youssef. Vertcal sem-symmetrc metrc connectons. Tensor, N.S., Vol. 49 (1990),