A Global Approach to Absolute Parallelism Geometry

Transcription

1 A Global Approac to Absolute Parallelsm Geometry Nabl L. Youssef 1,2 and Waleed A. Elsayed 1,2 arxv: v4 [gr-qc] 15 Jul Department of Matematcs, Faculty of Scence, Caro Unversty, Gza, Egypt 2 Center for Teoretcal Pyscs CTP), Brts Unversty n Egypt BUE) Emals: nlyoussef@sc.cu.edu.eg, nlyoussef2003@yaoo.fr waleedelsayed@sc.cu.edu.eg, waleed.a.elsayed@gmal.com Abstract. In ts paper we provde a global nvestgaton of te geometry of parallelzable manfolds or absolute parallelsm geometry) frequently used for applcaton. We dscuss te dfferent lnear connectons and curvature tensors from a global pont of vew. We gve an exstence and unqueness teorem for a remarable lnear connecton, called te canoncal connecton. Dfferent curvature tensors are expressed n a compact form n terms of te torson tensor of te canoncal connecton only. Usng te Banc denttes, some nterestng denttes are derved. An mportant specal fourt order tensor, wc we refer to as Wanas tensor, s globally defned and nvestgated. Fnally a double-vew for te fundamental geometrc obects of an absolute parallelsm space s establsed: Te expressons of tese geometrc obects are computed n te parallelzaton bass and are compared wt te correspondng local expressons n te natural bass. Pyscal aspects of some geometrc obects consdered are ponted out. Keywords: Absolute parallelsm geometry, Parallelzaton vector feld, Parallelzaton bass, Canoncal connecton, Dual connecton, Banc denttes, Wanas tensor. MSC 2010: 53C05, 53A40, 51P05. PACS 2010: Hw, Na, q, arxv: [gr-qc] 1

2 0 Introducton At te begnnng of te 20 t century, te mportance of geometry n pyscal applcatons as been llumnated by Albert Ensten. He as advocated a new plosopy nown as Te Geometerzaton Plosopy. Ts plosopy can be summarzed n te followng statement: To understand nature, one as to start wt geometry and end wt pyscs [15]. In 1915, Ensten used ts plosopy to understand te essence of Gravty, startng wt a 4-dmensonal Remannan geometry, endng wt a successful teory for gravty; te General teory of Relatvty GR) [3]. After te success of te teory, by testng ts predctons and applcatons, many autors ave drected ter attenton to te use of geometry to solve pyscal problems. Ensten n s contnuous attempts to understand more pyscal nteractons, as searced for a wder geometry to unfy gravty and electromagnetsm. Te problem wt Remannan geometry, owever, s tat t as only ten degrees of freedom te components of te metrc tensor n four dmensons) wc are ust suffcent to descrbe gravty. Tus to construct a successful geometrc teory tat would encompass bot gravty and electromagnetsm, one needs to enlarge te number of degrees of freedom. Ts canbedonentwo dfferent ways: eter byncreasng tedmenson of te underlyng spaceà la-kaluza-klen) or by replacng te Remannan structure by anoter geometrc structure avng more degrees of freedomwtout ncreasng te dmenson of te underlyng space). In taclng te problem of unfcaton, Ensten as cosen te second alternatve. Ts led m to consder Absolute Parallelsm geometry AP-geomety) [2] wc as sxteen degrees of freedom te number of components of te vector felds formng te parallelzaton); sx extra degrees of freedom are ganed. Many developments of AP-geometry ave been aceved e.g., [12, 14, 16]). Teores constructed n ts geometry e.g., [6, 7, 12]) togeter wt applcatons e.g., [8, 9, 13]) sow te advantages of usng AP-geometry n pyscs. Moreover, absolute parallelsm caracterzes te generalzed Berwald spaces among te Fnsler spaces [10, 11]. In ts paper, we establs a global approac to AP-geometry. Te global formulaton of te dfferent geometrc aspects of AP-geometry as many advantages. Some advantages of te global formalsm are: It could gve more nsgt nto te nfra-structure of pyscal teores constructed n te context of AP-geometry. Moreover, t may offer te opportunty to unfy feld teores n a more economc sceme. It elps better understand te meanng and te essence of te geometrc obects and formulae wtout beng trapped nto te complexty of ndces. As a consequence, t reduces te probablty of mstae It connects AP-geometry wt te modern language of te dfferental geometry. In local coordnates some mportant expressons, suc as te Le bracet [ x, x ], dsappear. Consequently, te contrbuton, geometrcal or pyscal, of all Le bracets are completely 2

3 dden. Suc expressons do not vans n global formalsm. Ts may produce new geometrc or pyscal nformaton. Te local formalsm represents rougly a mcro vewpont or a mcro approac wereas te global formalsm represents a macro vewpont. Te two vewponts are not alternatves but rater complementary and are ndspensable bot for geometry and pyscs. As global results old on te entre manfoldnot only on coordnate negboroods), tey also old locally. Te converse s not true; a result may old locally but not globally. Moreover, one can easly sft from global to local; t suffces to vew te global result n a coordnate negborood. Tese are te man motvatons of te present wor, were all results obtaned are formulated n a prospectve modern coordnate free form. Te paper s organzed n te followng manner. In secton 1, we defne globally te basc elements of te AP-geometry and prove an exstence and unqueness teorem for a remarable lnear connecton wc we call te canoncal connecton a flat) connecton for wc te parallelzaton vector felds are parallel ). We also study some propertes of ts connecton. In secton 2, we defne tree oter natural connectons te dual, symmetrc and Lev-Cvta connectons) and nvestgate ter propertes togeter wt te tensor felds assocated to tem. In secton 3, we express te curvature tensors of te above mentoned tree connectons n a smple and compact form n terms of te torson tensor of te canoncal connecton only. We ten use te Banc denttes to derve some furter nterestng denttes. In secton 4, we gve a global treatment of te W-tensor and nvestgate some of ts propertes. In secton 5, we present a double-vew for te fundamental geometrc obects of AP-geometry: On one and, we consder te local expressons of tese geometrc obects n te natural bass and, on te oter and, we compute ter expressons n te parallelzaton bass, and ten compare between te two sets of expressons. It sould fnally be noted tat ts wor s based manly on [16]. Trougout te present paper we use te followng notaton: M: an n-dmensonal smoot real manfold, FM): ter-algebra of C functons on M, XM): te FM)-module of vector felds on M, T x M: te tangent space to M at x M, T x M: te cotangent space to M at x M. We mae te assumpton tat all geometrc obects we consder are of class C. 1 Canoncal connecton In ts secton, we gve te defnton of an AP-space and prove an exstence and unqueness teorem for a remarable lnear connecton, wc we call te canoncal connecton. Also, we prove some propertes concernng ts connecton. 3

4 Defnton 1.1. [1] A parallelzable manfold s a par M, X), were M s an n-dmensonal smoot manfold and X = 1,..., n) are n ndependent vector felds defned globally on M. Te vector felds..., 1 n X are sad to form a parallelzaton on M. Suc a space s also nown n te lterature as an Absolute Parallelsm space or a Teleparallel space. For smplcty, we wll rater use te expressons AP-space and AP-geometry. Snce X are n ndependent vector felds on M, Xx) : = 1,..., n} s a bass of T x M for every x M. Any vector feld Y XM) can be wrtten globally as Y = Y were Y FM). Here we use te notaton Y to denote te components of Y wt respect to X. Ensten summaton conventon wll be appled on Latn ndces watever ter poston s even f te two repeated ndces are downword). Defnton 1.2. Te n dfferental 1-forms are called te parallelzaton forms. Clearly, f Y = Y ten ΩY) = Y, It follows drectly from 1.1) tat Ω x = Ω TxM : = 1,..., n} s te dual bass of te parallelza- Ω x : = 1,..., n} te dual parallelzaton ton bass Xx) : = 1,..., n} for every x M. We call Ω : XM) FM) defned by Ω X) = δ 1.1) ΩY) X = Y. 1.2) bass of T x M. Te parallelzaton forms Ω are ndependent n te FM)-module X M). Lemma 1.1. Let D be a lnear connecton on M. Te D-covarant dervatve of only f te D-covarant dervatve of X vanses. Proof. For every Y, Z XM), we ave, by 1.2) and 1.1), D Y Ω)Z) = D Y Ω) ΩZ) X ) = ΩZ) ΩD Y X). Ω vanses f and Consequently, by 1.2), from wc te result follows. DY Ω)Z) ) X = ΩZ)D Y Teorem 1.1. On an AP-space M, X), tere exsts a unque lnear connecton for wc te parallelzaton vector felds X are parallel. Proof. Frst we prove te unqueness. Assume tat s a lnear connecton satsfyng te condton X = 0. For all Y, Z XM), we ave Y Z = Y ΩZ) X ) = ΩZ) Y X + Y ) ΩZ) X = Y ) ΩZ) X. Hence, te connecton s unquely determned by te relaton Y Z = Y 4 ΩZ) ) X. 1.3)

5 To prove te exstence, let : XM) XM) XM) be defned by 1.3). We sow tat s a lnear connecton on M. In fact, let Y, Y 1, Y 2, Z, Z 1, Z 2 XM), f FM). It s clear tat Y1 +Y 2 Z = Y1 Z + Y2 Z and Y Z 1 +Z 2 ) = Y Z 1 + Y Z 2. Moreover, fy Z = fy) ) ΩZ) X = f Y ) ΩZ) X = f YZ, Y fz) = Y ΩfZ) ) X = Y f ΩZ) )) X = f Y ΩZ) ) X +Y f) ΩZ) X = f Y Z +Y f)z, by 1.2) and 1.3). It remans to sow tat satsfes te condton X = 0: Ts completes te proof. Y X = Y Ω X) ) X = Y δ ) X = 0. As a consequence of Lemma 1.1, we also ave Ω = 0. Hence X = 0, Ω = ) Ts property s nown locally) n te lterature as te AP-condton. Defnton 1.3. Let M, X) be an AP-space. Te unque lnear connecton on M defned by 1.3) wll be called te canoncal connecton of M, X). Te canoncal connecton s of crucal mportance because almost all geometrc obects n te AP-space wll be bult up of t, as wll be seen trougout te paper. Now we gve an ntrnsc formula of te torson tensor T of. Proposton 1.1. Te torson tensor T of te canoncal connecton s gven by TY, Z) = ΩY) ΩZ)[ X]. 1.5) Proof. Te torson tensor T of s defned, for all Y, Z XM), by Usng te AP-condton 1.4), we get TY, Z) TY, Z) = Y Z Z Y [Y, Z]. 1.2) = T ΩY) ΩZ) X ) = = ΩY) ΩZ) X X ΩY) ΩZ)T X) X X [ X]) = ΩY) ΩZ)[ X]. Teorem 1.2. Let M, X) be an AP-space. Te canoncal connecton of M, X) s flat. Proof. Te result follows from te defnton of te curvature tensor R of : and te AP-condton 1.4). RY, Z)V = Y Z V Z Y V [Y,Z] V 5

6 Remar 1.1. [16] It s for ts reason tat many autors tn tat te AP-space s a flat space. Ts s by no means true. In fact, t s meanngless to spea of curvature wtout reference to a connecton. All we can say s tat te AP-space s flat wt respect to ts canoncal connecton. However, tere are oter tree natural connectons on an AP-space wc are nonflat, as wll be sown later. 2 Oter lnear connectons on an AP-space In ts secton, we defne a metrc on an AP-space and nvestgate te propertes of tree oter natural connectons on te space. Moreover, we defne te contorton tensor and gve ts relaton to te torson tensor 1.5). Teorem 2.1. Let M, defnes a metrc tensor on M. X) be an AP-space and g := Ω te parallelzaton forms on M. Ten Ω Ω 2.1) Proof. Clearly g s a symmetrc tensor of type 0,2) on M. For all Y XM), we ave Moreover, gy,y) = 0 = gy, Y) = Ω Ω)Y,Y) = n ΩY) )2 0. =1 n ΩY) ) 2 = 0 = ΩY) = 0 = ΩY) X = 0 1.2) = Y = 0. =1 Hence, g s a metrc tensor on M. Remar 2.1. It s clear tat: a) g X) = δ. b) g Y) = ΩY). 2.2) 2.3) Property a) sows tat te parallelzaton vector felds X are g-ortonormal and b) provdes te dualty between Lemma 2.1. Let M, and only f X and Ω va g. X) be an AP-space. A lnear connecton D on M s a metrc connecton f ΩD V X)+ ΩD V X) = 0. Proof. By smple calculaton, usng 2.2) and 2.3), one can sow tat from wc te result follows. D V g) X) = ΩD V X) ΩD V X), 6

7 Te last lemma togeter wt te AP-condton 1.4) gve rse to te next result. Proposton 2.1. Te canoncal connecton s a metrc connecton. Proposton 2.2. On an AP-space tere are tree oter bult-n) lnear connectons: a) Te dual connecton gven by Y Z := Z Y +[Y,Z]. 2.4) b) Te symmetrc connecton gven by Y Z := 1 2 YZ + Z Y +[Y,Z]). 2.5) c) Te Lev-Cvta connecton s gven by [5] 2g Y Z, V) = Y gz, V)+Z gv, Y) V gy, Z) gy, [Z, V])+gZ, [V, Y])+gV, [Y, Z]). 2.6) Te proof s stragtforward and we omt t. Remar 2.2. One can easly sow tat: a) Y Z = Y Z TY, Z). b) Y Z = Y Z 1TY, Z) = YZ + Y Z). c) and are torsonless wereas and ave te same torson up to a sgn. Here T s te torson tensor of te canoncal connecton. Snce tere are no oter torson tensors n te space, we can say tat T s te torson of te space. In Remannan geometry te Lev-Cvta connecton as no explct expresson. However, n AP-geometry we can ave an explct expresson for te Lev-Cvta connecton as sown n te followng. Teorem 2.2. Let M, te form: X) be an AP-space. Ten te Lev-Cvta connecton can be wrtten n were L Y s te Le dervatve wt respect to Y XM). Y Z = Y Z 1 2 L X g)y, Z) 2.7) 7

8 Proof. By replacng V n 2.6) by 2Ω Y Z) = Y ΩZ)+Z X and usng 2.3), we get ΩY) Tang nto account 1.2) and 1.3), te above equaton reads X gy, Z)+gY, [ Z])+gZ, [ Y])+ Ω[Y, Z]). 2 Y Z = Y Z + Z Y X gy, Z)) X +gy, [ Z]) X +gz, [ Y]) X +[Y, Z] ) = 2 Y Z X gy, Z) gy, [ Z]) gz, [ Y]) by 2.5) = 2 Y Z LX g)y, Z) X. Corollary 2.1. In an AP-space, te Lev-Cvta connecton and te symmetrc connecton concde f, and only f, te parallelzaton vector felds are Kllng vector felds: = L X g = 0. Defnton 2.1. Te contorton tensor C s defned by te formula: Te contorton tensor may also be wrtten n te form: In fact, usng 1.2) and 1.3), we ave for all Y, Z XM), CY, Z) = Y Z Y Z. 2.8) CY, Z) = Y Ω)Z) X. 2.9) CY, Z) = Y Z Y Z = Y Ω Z) ) X Ω Y Z) X = Y Ω)Z) X. Te denttes 2.8) and 2.9) sow tat te geometry of an AP-space can be bult up from te Lev-Cvta connecton nstead of te canoncal connecton: Y Z = Y Z + Y Ω)Z) X. Te next proposton establses te mutual relatons between te torson and contorton tensors. Proposton 2.3. Te followng denttes old: a) TY, Z) = CY, Z) CZ, Y). ) b) CY, Z) = 1 TY, Z)+T Y, Z) X +T Z, Y) X. 2 From wc, a) TY, Z, V) = CY, Z, V) CZ, Y, V). ) b) CY, Z, V) = 1 TY, Z, V)+TV, Y, Z)+TV, Z, Y), 2 8

9 were CY, Z, V) = g CY, Z), V ) and TY, Z, V) = g TY, Z), V ). Consequently, te torson tensor vanses f and only f te contorton tensor vanses. Proof. Let Y, Z, V XM). Ten, a) Te frst dentty gves te torson tensor n terms of te contorton tensor. TY, Z) = Y Z Z Y [Y, Z] = Y Z Z Y) Y Z Z Y), snce s torsonless = Y Z Y Z) Z Y Z Y) = CY, Z) CZ, Y). b) Te second dentty gves te contorton tensor n terms of te torson tensor. In te followng proof we mae use of 2.7), Remar 2.2, 1.2), Remar 2.1 and 1.5). 2CY, Z) = 2 Y Z 2 Y Z = 2 Y Z 2 Y Z +LX g)y, Z) X = 2 Y Z 2 Y Z +TY, Z)+ ΩY) ΩZ)L X g) X) X = TY, Z)+ ΩY) ΩZ) X g X) g[ X], X) g[ = TY, Z) g TY, X), Z ) +g TZ, X), Y )) X ) = TY, Z)+ T Y, Z)+T Z, Y) X. X], X) ) X Remar 2.3. TY, Z, V) s sew-symmetrc n te frst two arguments wereas CY, Z, V) s sew-symmetrc n te last two arguments. Defnton 2.2. Let M, for every Y XM) by X) be anap-space. Te contracted torson or te basc formb s defned, BY) := TrZ TZ, Y)}. Ts 1-form s nown locally) n te lterature as te basc vector. In terms of te metrc tensor 2.1), usng 2.2), te basc form can be wrtten as BY) = g T Y), X ) = T Y, X). 2.10) Usng Proposton 2.3b), BY) can also be expressed n te form Mang use of 2.10) and 1.5), we ave BY) = C Y, X). BY) = ΩY) Ω[ X]). It sould be noted tat n te above tree expressons and n smlar expressons summaton s carred out on repeated mes ndces, altoug tey are stuated n dfferent argument postons. 9

10 Proposton 2.4. Concernng te four connectons of te AP-space, te dfference tensors are gven by: a) Y Z Y Z = TY, Z). b) Y Z Y Z = 1 TY, Z). 2 c) Y Z Y Z = CY, Z). d) Y Z Y Z = 1 TY, Z). 2 e) Y Z Y Z = CZ, Y). f) Y Z Y Z = 1L 2 X g)y, Z) X. Proof. Propertes a), b), d) follow from Remar 2.2, c) s te defnton of te contorton tensor, e) follows from 2.8) and te fact tat s torsonless, and f) follows from 2.7). As a consequence of te above proposton, we ave te followng useful relatons. Corollary 2.2. For every Y, Z, V XM), we ave te followng relatons: a) V T)Y, Z) V T)Y, Z) = S T V, TY, Z) )}. b) V T)Y, Z) V T)Y, Z) = 1 2 S T V, TY, Z) )}. c) V T)Y, Z) V T)Y, Z) = T Y, CV, Z) ) +T Z, CV, Y) ) +C V, TY, Z) ), were S denotes te cyclc permutaton of Y, Z, V and summaton. 3 Curvature tensors and Banc denttes In an AP-space te curvature R of te canoncal connecton vanses dentcally. Ts secton s devoted to sow tat te oter tree curvature tensors R, R and R, assocated wt, and respectvely, do not vans. Also, we sow tat te vansng of R enables us to express tese tree curvature tensors n terms of te torson tensor only. Teorem 3.1. Te tree curvature tensors R, R and R of te connectons, and are gven respectvely by: a) RY, Z)V = V T)Y, Z). 3.1) 10

11 b) 1 RY, Z)V = Z T)Y, V) Y T)Z, V) ) T TY, Z), V ) + 1 T Y, TZ, V) ) T Z, TY, V) )). 3.2) 4 c) RY, Z)V = Z C)Y, V) Y C)Z, V) C TY, Z), V ) +C Y, CZ, V) ) C Z, CY, V) ). 3.3) Proof. We prove a) only. Te proof of te oter parts can be carred out n te same manner. Usng 2.4), we get Y Z V = Y V Z +[Z, V]) = Y V Z + Y [Z, V] = V ZY +[Y, V Z]+ [Z,V] Y +[Y, [Z, V]]. Smlarly, and Z Y V = V YZ +[Z, V Y]+ [Y,V] Z +[Z, [Y, V]]. [Y,Z] V = V [Y, Z]+[[Y, Z], V]. Usng te above tree denttes, togeter wt te Jacob dentty, we get RY, Z)V = Y Z V Z Y V [Y,Z] V = V ZY +[Y, V Z] V YZ [Z, V Y] V [Y, Z]+ [Z,V] Y [Y,V] Z. Usng te fact tat te curvature tensor of te canoncal connecton vanses Teorem 1.2), t follows tat [Y,Z] V = Y Z V Z Y V. Usng te above dentty, we get RY, Z)V = V ZY +[Y, V Z] V YZ [Z, V Y] V [Y, Z] + Z V Y V Z Y Y V Z + V Y Z = V Y Z V Z Y V [Y, Z]) V YZ Z V Y [ V Y, Z]) Y V Z V ZY [Y, V Z]) = V TY, Z) T V Y, Z) TY, V Z) = V T)Y, Z). Te above teorem sows tat te curvature tensors R, R and R are expressble n terms of te torson tensor of te space only. Ts proves tat te geometry of an AP-space depends crucally on te torson tensor. It s wort mentonng tat te vansng of tat tensor mples tat te 11

12 four connectons,, and concde and a trval flat Remannan space s aceved. Tus, a suffcent condton for te non-vansng of te torson tensor s te non-vansng of any one of te tree curvature tensors R, R or R. Te next result gves te expressons of te Rcc tensor Rc of and te Rcc-le tensors Rc and Rc of and togeter wt ter respectve contractons te scalar curvature Sc and te curvature-le scalars Sc and Ŝc). Te ortonormalty of te parallelzaton vector felds X plays an essental role n te proof. Teorem 3.2. In an AP-space M, a) RcY, Z) = Z B)Y). b) RcY, Z) = 1L 2 X T)Y, Z, X)+ 1T Y, TZ, 4 X) we ave, for every Y, Z XM), X), X ) 1 2 YB)Z) 1B TY, Z) ). 4 c) RcY, Z) = LX C)Y, Z, X)+CY, C Z, X), X ) Y B)Z) BCY, Z)). a) Sc = X B X). b) Ŝc = 1 X B X)+ 1T [ 2 4 X], X). c) Sc = 2 X B X)+B X)B X)+CT X), X)+C C X), Proof. We prove b) and c) only. Te oter denttes can be proved smlarly. b) Usng 2.2), 3.2), 1.4) and 2.10), we ave X ). RcY, Z) = g RY, X)Z, X ) = 1 2 X T)Y, Z, X) Y T) Z, X) T TY, X), Z, X )) + 1 T Y, T Z), X ) T TY, Z), X )) 4 = 1 2X TY, Z, X) 2T 4 X Y, Z, X) 2TY, X Z, X) 2 Y B)Z) 2T TY, X), Z, X ) +T Y, T Z), X ) B TY, Z) )) = 1 2X TY, Z, X) TY, 4 X Z, X) 2 Y B)Z) 2T[ Y], Z, X) TY, [ Z], X) B TY, Z) )) = 1 2L 4 X T)Y, Z, X) TY, X Z, X) TY, [Z, X], X) 2 Y B)Z) B TY, Z) )) = 1 2 L X T)Y, Z, X)+ 1 4 T Y, TZ, X), X ) 1 2 YB)Z) 1 4 B TY, Z) ). 12

13 c) Usng 2.2), c), 1.4), 2.8), Proposton 2.3b) and te torsonless property of, we get Sc = Rc X) X C) = L X)+C C X), X ) X B) X) B C = X C X) C[ X], X) C [ X], X) X B X) C X X)+B X)B X) = 2X B X)+B X)B X) C[ X], ) C X X)+C [ X], X) = 2X B X)+B X)B X)+C T X), X ) C X X ) = 2X B X)+B X)B X)+C T X ) +C C X), X) X), X) ) X ). Table1: Lnear connectons n AP-geometry Connecton Symbol Torson Curvature Metrcty Canoncal T 0 metrc Dual T R nonmetrc Symmetrc 0 R nonmetrc Lev-Cvta 0 R metrc Let D beanarbtrarylnear connecton onm wt torsontandcurvature R. Ten tebanc denttes are gven, for all Y, Z, V, U XM), by [4]: } Frst Banc dentty: S RY, Z)V = S D V T)Y, Z)+T TY, Z), V )}. Second Banc dentty: S D V R)Y, Z)U R V, TY, Z) ) U } = 0. In wat follows, we derve some denttes usng te above Banc denttes. Some of te derved denttes wll be used to smplfy oter formulae tus obtaned. Proposton 3.1. Te frst Banc dentty for te connectons,, and reads: a) S V T)Y, Z)+T TY, Z), V )} = 0. 13

14 } b) S RY, Z)V = S T TY, Z), V ) } V T)Y, Z). } c) S RY, Z)V = 0. RY, } d) S Z)V = 0. Te proof s stragtforward. We ave to use te relatons R = 0, T = T and T = T = 0. Corollary 3.1. Te followng denttes old: } a) S V T)Y, Z) = 2 S T TY, Z), V )}. b) S V T)Y, Z) } = 1 S T TY, Z), V )}. 2 } c) S RY, Z)V = S T TY, Z), V )}. Te proof follows from te above proposton togeter wt Corollary 2.2 and 3.1). Proposton 3.2. Te second Banc dentty for te connectons, and reads: } a) S V R)Y, Z)U = S U T) TY, Z), V )}. } b) S V R)Y, Z)U = 0. c) S } VR)Y, Z)U = 0. Te proof s stragtforward mang use of 3.1). Now, we wll gve anoter formula for te curvature tensor R of te symmetrc connecton wc s more compact tan 3.2). Teorem 3.3. Te curvature tensor R can be wrtten n te form: RY, Z)V = 1 2 VT)Y, Z) 1 4 Proof. Tang nto account 3.2) and Proposton 3.1a), one as RY, Z)V = = 1 4 S S S = 1 2 VT)Y, Z) 1 4 T Y, TZ, V) ) +T Z, TV, Y) )). } V T)Y, Z) VT)Y, Z) T V, TY, Z) )} T V, TY, Z) ) T V, TY, Z) )} T V, TY, Z) ) VT)Y, Z) T Y, TZ, V) ) +T Z, TV, Y) )). 14

15 Corollary 3.2. On an AP-space M, connecton can be wrtten as: X) te Rcc-le tensor Rc wt respect to te symmetrc RcY, Z) = 1 2 ZB)Y)+ 1 4 B TY, Z) ) 1 4 T Y, T Z), X ). It s to be noted tat te expresson S T TY, Z), V )} appears n many of te denttes obtaned above. We dscuss now te case n wc ts expresson vanses. Let us wrte [ X] =: C X. Te functons C FM) are global functons on M and wll be referred to as te global structure coeffcents of te AP-space. Tey can be wrtten explctly n te form C = structure coeffcents. S Ω[ X]). Te last expresson may be consdered as a defnton of te global Teorem 3.4. On an AP-space M, X) te expresson S T TY, Z), V )} vanses f and only } f, for all, te expresson S C,, X vanses. Consequently, f te global structure coeffcents of te AP-space are constant functons on M, ten T TY, Z), V )} = 0. Proof. Usng te parallelzaton vector felds nstead of Y, Z and V, we ave: S T T X), X )} } = 0 S T[ X], X) = 0, by 1.5),,,, S,, X [ X]+ [ [ X], X ]} = 0, by 1.4) } S,, X [ X] = 0, by Jacob dentty S X,, Ω[ X]) ) X S,, X Ω[ X]))} X = 0 S X Ω[,, } S C,, X } = 0, by 1.3) X]) )} = 0, by te ndependence of = 0, by 1.2). It sould be noted tat for te natural bass : α = 1,..., n}, te bracet [, ] = 0 and x α x α x β so te structure coeffcents assocated wt ) vans. For ts reason te local expresson n x α te natural bass) of te dentty S T TY, Z), V )} = 0 s vald as s establsed n [16]. Te last proposton gves rse to te followng nterestng formulae. Corollary 3.3. In an AP-space M, X), f te global structure coeffcent of te AP-space are constant functons on M, ten te next formulae old: a) V T)Y, Z) = V T)Y, Z) = V T)Y, Z). } b) S V T)Y, Z) = X

16 } c) S V T)Y, Z) = 0. d) S V T)Y, Z) } = 0. } e) S RY, Z)V = 0. f) RY, Z)V = 1 2 VT)Y, Z) 1T TY, Z), V ). 4 } } ) S V C)Y, Z) = S V C)Z, Y). 4 Wanas Tensor Te Wanas tensor was frst defned locally by M. I. Wanas n 1975 [12]. It as been used by F. Mal and M. Wanas [6] to construct a pure geometrc teory unfyng gravty and electromagnetsm. In ts secton, we ntroduce te global defnton of te Wanas tensor and nvestgate t. Defnton 4.1. Let M, te formula X) be an AP-space. Te tensor feld W of type 1,3) on M defned by WY, Z) X = 2 Y,ZX 2 Z,Y were 2 Y,Z = Y Z Y Z, s called te Wanas tensor, or smply te W-tensor, of M, X). Usng 1.2), for every Y, Z, V XM), we get WY, Z)V = 2 Y,Z X 2 Z,Y X ) ΩV). 4.1) Te next result gves a qute smple expresson for a suc tensor. Teorem 4.1. Te W-tensor satsfes te followng dentty WY, Z)V = RY, Z)V T TY, Z), V ). 4.2) Proof. Consder te commutaton formula for te parallelzaton vector feld Consequently, WY, Z)V 4.1) = 2 Y,ZX 2 Z,Y X = RY, Z) X TY,Z) X ΩV) RY, Z) X ΩV) TY,Z) X = RY, Z)V + ΩV) TY,Z) by 1.2) = RY, Z)V + TY,Z) ΩV) X TY, Z) ΩV) ) X = RY, Z)V + TY,Z) V TY,Z) V, by 1.2) and 1.3) = RY, Z)V T TY, Z), V ), by Proposton X wt respect to :

17 Corollary 4.1. Te W-tensor can be expressed n te form: WY, Z)V = V T)Y, Z) T TY, Z), V ). 4.3) In fact, ts expresson follows from 3.1). Ts sows tat te W-tensor s expressed n terms of te torson tensor of te AP-space only. Proposton 4.1. Te Wanas tensor as te followng propertes: a) WY, Z)V s sew symmetrc n te frst two arguments Y, Z. } b) S WY, Z)V = 2 S T TY, Z), V )}. c) S WY, Z)V } } = S V T)Y, Z). Proof. Property a) s trval, b) follows from Proposton 3.1a) and 4.3), c) follows from b) and Corollary 3.1a). Te dentty satsfed by te W-tensor n Proposton 4.1b) s te same as te frst Banc dentty Corollary3.1c) ) oftedualcurvaturetensoruptoaconstant. Tedenttycorrespondng to te second Banc dentty s gven by: Proposton 4.2. Te W-tensor satsfes te followng dentty: } S V W)Y, Z)U = S T T TY, Z), V ) ), U +T T TY, Z), U ), V )+T TU, V), TY, Z) )} + S U T) TY, Z), V ) V T) TY, Z), U )} 4.4) Proof. Tang nto account 4.2) togeter wt Proposton 3.2a), Corollary 3.1a) and Corollary 2.2a), we get S = S } V W)Y, Z)U V R)Y, Z)U V T) TY, Z), U ) T V T)Y, Z), U ))} = S U T) TY, Z), V ) V T) TY, Z), U ) 2T = S S 2T = S S T U T) TY, Z), V ) V T) TY, Z), U )} T TY, Z), V ) ), U T TY, Z), V ) )}, U + S T T TY, Z), U ) )}, V V,TY,Z),U U T) TY, Z), V ) V T) TY, Z), U )} T TY, Z), V ) ), U Corollary 4.2. In an AP-space M, constant, we ave +T T TY, Z), U ) ), V +T TU, V), TY, Z) )}. X), f te global structure coeffcent of te AP-space are 17

18 } a) S WY, Z)V = 0. b) S V W)Y, Z)U } = S U T) TY, Z), V ) V T) TY, Z), U )} Te proof s stragtforward from Teorem 3.4 and Corollary 3.3. We end ts secton by te followng comments and remars on Wanas tensor. Te W-tensor s defned by usng te commutaton formula wt respect to te dual connecton. Notng new arose from te same defnton f we use te tree oter connectons, and ). Usng te commutaton formula for te parallelzaton form vector feld X n te defnton of te W-tensor: WY, Z)V = 2 Z,Y Ω)V) 2 Y,Z Ω)V) Ω nstead of te parallelzaton ) X gves te same formula 4.2) for te W-tensor and consequently te same propertes. Beng defned by usng te parallelzaton vector felds te Wanas tensor s defned only n AP-geometry. It as no analogue n oter geometres. Altoug te W-tensor and te dual curvature tensor ave some common propertes for example, Proposton 4.1b)), tere are sgnfcantly dfferent propertes for example, 4.4)). In te case of constant global structure coeffcents, te W-tensor as some propertes common wt te Remannan curvature R. For a pyscal dscusson concernng te W-tensor we refer to [16]. 5 Parallelzaton bass versus natural bass Ts secton s devoted to a double-vew for te fundamental geometrc obects of AP-geometry. On one and, we consder te local expressons of tese geometrc obects n te natural bass [16] and, on te oter and, we compute ter expressons n te parallelzaton bass, gvng rse to a concse table expressng ts double-vew. Let U, x α ) ) bealocal coordnatesystem ofm. At eacpont x U, we avetwo dstngused bases of T x M, namely, te natural bass µ := : µ = 1,..., n} and te parallelzaton bass x µ Xx) : = 1,..., n}. Tese two bases are fundamentally dfferent. Te parallelzaton vector felds X are defned globally on te manfold M wereas te natural bass vector felds µ are defned only on te coordnate negborood U. Consequently, te natural bass vector felds depend crucally on coordnate systems wereas te parallelzaton vector felds do not. Gree world) ndces are related to te natural bass and Latn mes) ndces are related to te parallelzaton bass. Ensten summaton conventon wll be appled as usual on Gree ndces. 18

19 It wll also be appled on Latn ndces watever ter poston s even f te two repeated ndces are upward or downward). A tensor feld H of type r, s) on M s wrtten n te natural bass n te form: and n te parallelzaton bass n te form: were H α 1...α r µ 1...µ s FU) and H 1... r 1... s FM). H = H α 1...α r µ 1...µ s α1... αr dx µ 1... dx µs, on U H = H 1... r 1... s X 1... X r Ω 1... Ω s, on M, A vector feld Y XM) s wrtten n te natural bass n te form Y = Y α α and n te parallelzaton bass n te form Y = Y X. In partcular, X = X α α and α = Ω α ) X = Hence X α ) 1 α, n s te matrx of cange of bases and Ω α ) 1 α, n s te nverse matrx. We use te followng notatons wt smlar notatons wt respect to mes ndces): Γ α µν, Γ α µν, Γ α µν, Γ α µν : te coeffcents of te lnear connectons,,, respectvely, : te covarant dervatve wt respect to te dual connecton, g µν resp. g µν ): te covarant resp. contravarant) components of te metrc tensor g, Λ α µν : te components of te torson tensor T, B α : te components of te basc form B, γµν α : te components of te contorton tensor C, Wσµν: α te components of te Wanas tensor W. Ω α X. Let D be an arbtrary connecton on M wt torson tensor T and curvature tensor R. We use te followng conventons: D µ ν = D α νµ α, T µ, ν ) = T α νµ α, R µ, ν ) σ = R α σµν α, wt smlar conventons wt respect to Latn ndces. Te next table gves a comparson between te most mportant geometrc obects of AP-geometry expressed n te natural bass and n te parallelzaton bass. Geometrc obects, equatons or denttes avng te same form n te two bases are not ncluded n tat table. However, f a geometrc obect as te same form n te two bases, tat s, f ts expressons n world ndces and mes ndces are smlar, ts does not mean tat te geometrc meanng of tese two expressons s te same. 19

20 Table2: Parallelzaton bass versus natural bass Geometrc obect Local form In te natural bass world ndces) Global form In te parallelzaton bass mes ndces) Parallelzaton vector felds, parallelzaton forms X α Ω α = δ, X α Ω µ = δµ α X = δ, Ω = δ Metrc tensor g µν = Ω µ Ω ν g = δ Canoncal connecton Γ α νµ = X α Ω ν,µ were, µ denotes µ Γ = 0 Dual connecton Γα νµ = Γ α µν Γ = C Symmetrc connecton Γα νµ = 1 2 Γα νµ +Γα µν ) Γ = 1 2 C Lev-Cvta connecton Γ α νµ = 1 2 gασ g σν,µ +g µσ,ν g νµ,σ ) ) Γ = 1 C 2 +C +C Torson tensor Λ α νµ = Γ α νµ Γ α µν Λ = C Contorton tensor γ α νµ = Γ α νµ Γ α νµ γ = Γ Torson n terms of contorton Contorton n terms of torson Λ α νµ = γα νµ γα µν Λ σνµ = γ σνµ γ σµν were Λ µνσ = g ǫµ Λ ǫ νσ and γ µνσ = g ǫµ γνσ ǫ ) γνµ α = Λ 1 ανµ +Λ 2 µνǫ +Λ νµǫ )g αǫ γ µνσ = 1 Λ 2 σνµ +Λ µνσ +Λ νσµ ) Λ = γ γ γ = 1 2 C +C +C ) Basc form B µ = Λ α µα = γα µα B = Λ = γ = C Wanas tensor Wσµν α = X α νµ X α µν ) Ω σ W = X X W α σµν = Λ ǫ µνλ α σǫ Λ α µν σ W = Cl Cl X C 20

21 Te above table merts some comments. We conclude ts secton and te paper by te followng remars and comments. Te trd column of te above table s obtaned by computng te expresson of te geometrc obects n te parallelzaton bass. For example, to compute te coeffcents of te Lev-Cvta connecton Γ, set Y = Z = V = X n 2.6). Ten, we get 2g X X) = X g X)+ X g g [ X])+g [ X) X g X])+g X) [ X]). For te left-and sde LHS), As X g vans. Hence, X) = LHS = 2g Γ l l X g = X) = 2g lγ l = 2δ l Γ l = 2 Γ. X δ = 0, te frst tree terms of te rgt-and sde RHS) RHS = g C l X)+g C l l X)+g C l l X) l = g l C l +g l C l +g l C l = C +C +C. Accordngly, Γ = 1 2 C +C C ). It s clear from te trd column tat almost all geometrc obects of AP-geometry are expressed n terms of te global structure coeffcents C. Te global structure coeffcents tus play a domnant role n AP-geometry formulated n mes ndces. Its role s smlar to, and even more mportant tan, te role played by te torson tensor Λ n AP-geometry formulated n world ndces. Te structure coeffcents C α µν wt respect to an arbtrary bass e α ) are not te components of a 1,2)-tensor feld. In fact, let e α = A α α e α under a cange of local coordnates from x α ) to x α ) and let [e µ,e ν ] = Cµνe α α and [e µ, e ν ] = Cµ α ν e α. Ten, one can easly sow tat te transformaton formula for Cµν α as te form: C α µ ν = Aα α Aµ µ A ν ν Cα µν +Kα µ ν Kα ν µ, were Kµν α = Aµ µ Aα α e µ Aα ν ). Tus, Cα µν are not te components of a tensor feld of type 1,2) unless e µ A α ν = 0 tat s, te matrx of cange of bases A α α s a constant matrx) or Kµν α s symmetrc wt respect to µ and ν. Also, te global structure coeffcents C are not te components of a 1,2)-tensor feld tey are n 3 functons defned globally on M and avng certan propertes). Neverteless, for fxed and, C are te components of te 1,0)-tensor feld [ X]. 21

22 In te parallelzaton bass, altoug te coeffcents of te canoncal connecton vans: Γ = 0 X X = 0 because of te AP-condton), ts torson tensor T does not vans: Λ = C ; a penomenon tat never exst n natural local coordnates. Ts s due to te non-vansng of te bracet [ X] n te expresson of te torson tensor: T X) = X X X X [ X]. For te same reason te dual connecton as also non-vansng coeffcents: Γ = C. From te table we ave Γ = 2 Γ = C and Γ = γ = 1 2 C +C +C ). Ts means tat te dual connecton coeffcents and te symmetrc connecton coeffcents concde up to a constant) and are bot equal to te global structure coeffcents up to a constant). On te oter and, te Lev-Cvta connecton coeffcents concde wt te contorton coeffcents up to a sgn). Ts sows agan tat everytng n AP-geometry s expressble n terms of te global structure coeffcents. Also, all survvng connectons n te space may be represented by only one of tem, say te Lev-Cvta connecton. A quc loo at te trd column of te above table may deceve and lead to erroneous conclusons: te symmetrc connecton s sew-symmetrc and te Lev-Cvta connecton s non-symmetrc. Ts s by no means true. Te formulaton of te noton of symmetry of connectons usng ndces s not appled any more n ts context. In fact, a lnear connecton s symmetrc f and only f t concdes wt ts dual connecton, and ts s te case for bot te symmetrc and Lev-Cvta connectons. Anoter example: altoug te symmetrc and dual connectons concde up to a constant) n te parallelzaton bass, te symmetrc connecton as no torson wle te dual connecton as a survvng torson. Ts s, once more, due to te fact tat te torson expresson as a bracet term wc does not depend on te connecton. Te torson and contorton tensors of type 0,3) are present n te natural bass wle tey are not n te parallelzaton bass. Ts s because te metrc matrx s te dentty matrx δ ). Consequently, mes ndces can not be rased or lowered usng te metrc g. In local coordnates te structure coeffcents vans: [ µ, ν ] = 0 wle n te parallelzaton bass te global structure coeffcents are alve: [ X] = C X). Fortsreasontestructure coeffcents, n local coordnates, ave no effect and te second column of te above table gve tus te usual expressons we are accustomed to [16]. As an example, as te connecton coeffcents depend on coordnate systems, te canoncal connecton coeffcents do not vans n te natural bass wle tey vans n te parallelzaton bass). For pyscal applcatons, especally n general relatvty and gravtaton, one can assgn a sgnature to te postve defnte metrc g defned by 2.1). Ts can be aceved, for n = 4, by wrtng g = η Ω Ω, were η = 0 for, η = 1 for = = 0, η = +1 for = = 1,2,3. Te metrc g s tus nomore postve defnte but rater nondegenerate. 22

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