Infinitesimal Rigidity and Flexibility at Non Symmetric Affine Connection Space

Transcription

1 ESI The Erwn Schrödnger Internatonal Boltzmanngasse 9 Insttute for Mathematcal Physcs A-9 Wen, Austra Infntesmal Rgdty and Flexblty at Non Symmetrc Affne Connecton Sace Ljubca S. Velmrovć Svetslav M. Mnčć Mća S. Stankovć Venna, Prernt ESI November, 6 Suorted by the Austran Federal Mnstry of Educaton, Scence and Culture Avalable va htt://

2 INFINITESIMAL RIGIDITY AND FLEXIBILITY AT NON-SYMMETRIC AFFINE CONNECTION SPACE Ljubca S. Velmrovć, Svetslav M. Mnčć, Mća S. Stankovć Faculty of Scence and Mathematcs Všegradska, 8 Nš Serba Abstract At the resent work we consder nfntesmal deformatons f : x x + εz x j of a sace L N wth non-symmetrc affne connecton L. Based on the non-symmetry of the connecton, we use four knds of covarant dervatve to exress the Le dervatve and the deformatons. Rgdty of geometrc objects connecton, tensors, curvature s defned by vrtue of Le dervatve. AMS Subj. Class.: 5C5, 5A5, 5B5 Key words: Le dervatve, rgdty, flexblty, non-symmetrc affne connecton sace, nfntesmal deformaton, curvature tensors Introducton Let us consder a sace L N of non-symmetrc affne connecton L wth the torson tensor T = L L kj, at local coordnates x =,..., N. Noton of non-symmetrc affne connecton s used for the frst tme, for examle, n [Es] Esenhart, 97, [Hy] Hayden, 9, but use of non-symmetrc connecton became esecally actual after aearance of the works of Ensten, relatng to create the Unfed Feld Theory UFT. Ensten was not satsfed wth hs General Theory of Relatvty GTR, 96, and from 9. to the end of hs lfe 955, he worked on varous varants of UFT. Ths theory had the am to unte the gravtaton theory, to whch s related GTR, and the theory of electromagnetsm.

3 L. S. Velmrovć, S. M. Mnčć, M. S. Stankovć Introducng dfferent varants of hs UFT, Ensten [En], 95, [En], 96 uses a comlex basc tensor g j, wth symmetrc real art and antsymmetrc magnary art wth resect to, j. Begnnng wth 95, n the work [En] Ensten uses real non-symmetrc basc tensor. At that sense are also hs works from ths feld u to the end of hs lfe, and also the last work [En5], 955. Remark that at UFT the symmetrc art g j of the basc tensor g j s related to gravtaton, and antsymmetrc g j to the electromagnetsm. The same s vald for Γ and Γ. Whle at the Remannan sace the sace of GTR the connecton coeffcents are exressed by vrtue of g j, at Ensten s works from UFT the connecton between these magntudes s determned by equatons g j ;m g j,m Γ m g j Γ mj g =, g j,m = g j + x m whch s a system of = 6 equatons wth 6 unknowns. Begnnng wth 95. L. P. Esenhart was n several works occued wth roblems of saces wth non-symmetrc basc tensor and non-symmetrc connecton. In the work [Es], 95, Esenhart defnes a generalzed Remannan sace GR N, as a sace of coordnates x =,..., N wth whch s assocated a non-symmetrc tensor g j, and the connecton coeffcents are defned by equatons Γ. = g j,k g, + g k,j, Γ = g Γ.. In the work [Es], n 95., Esenhart obtans two curvature tensors at GR N, usng the fact that connecton coeffcents are non-symmetrc. As t s known, one can assgn on dfferentable manfold connecton coeffcents L x,..., x N ndeendently of basc tensor, so that generally s L x L kjx and then we have a non-symmetrc affne connecton sace L N. There are dfferent defntons of the noton of nfntesmal deformatons. For examle, K. Yano [Yano78] a deformed magntude Ā x observes transmtted arallel from the from the ont x, obtaned on the base of the transformaton, at orgnal ont x and formes the dfference between the obtaned magntudes at x. At nfntesmal deformatons one observes unchangeablty of several geometrc magntudes,.e. one looks for the condtons to be Ā = A. When Ā = A, we say that the magntude A s unchangeable at nfntesmal deformaton,.e. the mentoned magntude s rgd, on the contrary that magntude s non-rgd or flexble. In the case of the rgdty of the arc, one says that we have an nfntesmal bendng of a manfold, artculary of a surface n E. The case of bendng s secally mortant ntensvely s studed. In ths case one observes changes of

4 Infntesmal rgdty and flexblty... others magntudes, and then we say that they are rgd or flexble. For examle, the coeffcents of the frst quadrat form are rgd, and of the second one are flexble n the nfntesmal bendng of a surface. In ths matter there are many works from the geometers [K-F], [Ef], [Vek], [A.D.Al], [IIK-S], [IIK-S]. Le dervatve and nfntesmal deformatons At the begnnng we are gvng some basc facts accordng to [MVS], [RSt6], [VMS],[Mn7]. Defnton. A transformaton f : L N L N : x = x,..., x N x x = x,..., x N x, where. x = x + zxε, or n local coordnates. x = x + z x j ε,, j =,..., N, where ε s an nfntesmal, s called nfntesmal deformaton of a sace L N, determned by the vector feld z = z, whch s called nfntesmal deformaton feld.. We denote wth local coordnate system n whch the ont x s endowed wth coordnates x, and the ont x wth the coordnates x. We wll also ntroduce a new coordnate system, corresondng to the ont x = x new coordnates. x = x,.e. as new coordnates x of the ont x = x we choose old coordnates at the system of the ont x = x. Namely, at the system s x = x =. x, where = denotes equal accordng to... Defnton.. Coordnate transformaton we get based on unctual transformaton f : x x, gettng for the new coordnates of the ont x the old coordnates of ts transform x, s called draggng along unctual transformaton. New coordnates x = x of the ont x are called dragged along coordnates. In the case of nfntesmal deformaton. coordnate transformaton. x = x = x + z x,..., x N ε s called draggng along by z ε.

5 L. S. Velmrovć, S. M. Mnčć, M. S. Stankovć Let us consder a geometrc object A wth resect to the system at the ont x = x L N, denotng ths wth A, x. Defnton. The ont x s sad to be deformed ont of the ont x, f. holds. Geometrc object Ā, x s deformed object A, x wth resect to deformaton., f ts value at the system, at the ont x s equal to the value of the object A at the system at the ont x,.e. f. Ā, x = A, x. Remark.. In ths study of nfntesmal deformatons accordng to. quanttes of an order hgher then the frst wth resect to ε are neglected. We wll now defne some mortant notons of the theory of nfntesmal deformatons, followng from.: Le dfferental and Le dervatve, and n further consderatons we wll fnd them for some geometrc objects. Defnton. The magntude DA, the dfference between deformed object Ā and ntal object A at the same coordnate system and at the same ont wth resect to.,.e..5 DA = Ā, x A, x, s called Le dfference Le dfferental, and the magntude.5 DA Ā, x A, x L z A = lm = lm ε ε ε ε s Le dervatve of geometrc object A, x wth resect to the vector feld z = z x j. Usng the relaton.5, for deformed object Ā, x we have.5 Ā, x = A, x + DA, and thus we can exress Ā, fndng revously DA. Defnton.5 Geometrc object A s rgd wth resect to nfntesmal deformatons. f there exst nontrval feld of deformatons z for whch L z A =,.e. Ā = A. Geometrc object A s flexble f L z A for non-trval feld of deformaton z. The known man cases are:.. Accordng to.5 we have Dx = x x,.e. for the coordnates we have.6 Dx = z x j ε,

6 Infntesmal rgdty and flexblty... 5 from where.6 L z x = z x j... For the scalar functon ϕx ϕx,..., x N we have.7 Dϕx = ϕ, z xε = L z ϕxε, ϕ, = ϕ/ x,.e. Le dervatve of the scalar functon s dervatve of ths functon n drecton of the vector feld z... For a tensor of the knd u, v we get.8 Dt...u = [t...u, z = L z t...u ε, α= z α, α t...u + z,j β β= t...u ]ε where we denoted.9 t...u = t...α α+...u, α t...u j...j v = t...u j...j β j β+...j v... For the vector dx we have. Ddx = L z dx =..5. In the same way, as for the tensors, for the connecton coeffcents we have. DL = L, z + z, z, L + z,j L k + z,k L j ε = L zl ε. Defnton.6 Infntesmal deformaton. s affne colneaton or rojectve nfntesmal deformaton f the Le dervatve of connecton coeffcents s zero L z L =. In case of nfntesmal deformaton wth zero Le dervatve of curvature LR = we have curvature colneaton. In the case of L z g j =, nfntesmal deformaton s nfntesmal moton, and for L z g j = σxg j, we have nfntesmal conformal deformaton or n the case σx = const.- homothety.

7 6 L. S. Velmrovć, S. M. Mnčć, M. S. Stankovć The Le dervatve and rgdty of a tensor.. Defnton. A tensor t...u s rgd wth resect to a gven nfntesmal deformaton feld z f L z t...u = Consderng of the rgdty of a tensor requres consderaton of ther Le dervatves. Based on non-symmetry of the connecton, at L N we can consder two tyes of covarant dervatves for a vector and four tyes for general tensor. So, denotng by =,..., dervatve of the tye, we have [Mn7], [Mn77], [Mn79]:.a d t...u m = t...u j +...j v,m α= L α m m m m Generally, the next theorem s n the force: α t...u β= L j βm mj β mj β j β m t...u j...j v. Theorem. Le dervatve of a tensor t...u of the tye u, v s a tensor of the same tye and can be resented n the followng four ways.a, b L z t...u = Lzt...u j...j v t...u z + =, ; α= α= T α s z α s α α t β= t......z + z j β β= T s j β t s t......z,.a, b L z t...u = Lzt...u j...j v t...u z + α= α= T α s z α s α α t β= t......z, =,, z j β t where L z denotes that Le dervatve L z s exressed by covarant dervatve of the tye,, =,...,. The roof s gven n [VMS ]. Corollary.. For the sace L N of symmetrc connecton L T =

8 Infntesmal rgdty and flexblty... 7 we have. L z t...u = L z t...u t...u ; z α= z α ; α t...u + z ;j β β= t...u, because n that case all tyes of covarant dervatves reduce to one, whch we denote by semcolon ;... So, Le dervatve of a tensor at a sace of symmetrc affne connecton can be obtaned as a secal case from the formulae for Le dervatve at a sace of non-symmetrc affne connecton. We wll nvestgate the way of resentng the Le dervatve of a tensor by covarant dervatve wth resect to symmetrcal art L of non-symmetrc connecton L. Let us consder a sace L N of non-symmetrc affne connecton L and let be.5 L = L + L kj, T = L L kj. Then.6 L = L + T. The magntudes L are the coeffcents of symmetrc connecton assocated to the connecton L, and T are the comonents of torson tensor of connecton L. If we denote wth L z t...u the exresson as on the rght sde at., but formed by means of L from.5 nstead of L, we have the next theorem [VMS ]: Theorem. In the non-symmetrc connecton sace L N Le dervatve of tensor t...u can be exressed as.7 L z t...u = Lzt...u j...j v = L zt...u t...u ; z α= z α ; α t...u + z ;j β β= t...u, where the semcolon ; denotes covarant dervatve wth resect to symmetrc art L of the connecton L.. Comarng., and.7, we can see that Le dervatve of a tensor at L N can smler be gven by means of.7,.e. wth resect to covarant dervatve formed by symmetrcal art L of non-symmetrcal connecton L.

9 8 L. S. Velmrovć, S. M. Mnčć, M. S. Stankovć If we use at the same tme dfferent knds of covarant dervatve at the rght sde at., wth resect to L, we can wrte ths equatons n the more condensed form analogously to.7. In connecton wth ths the next theorem s n the force: Theorem. The Le dervatve of a tensor of the tye u, v can be exressed usng covarant dervatves wth resect to non-symmetrc connecton L n the next way.8a d L z t...u = t...u z α z λ α= where λ, µ, ν {,,,,,,,,,,,}. µ α t...u + β= z νj β t...u, Proof: We wll rove only the second case, the others can be roved analogously. Let us start from.b. We have z α t...u j...j v = z, α + Lα s zs t...u j α...j v = z, α + Lα s zs L α s zs α + L α sz s and analogously α z j β t...u = z α α t...u + T α s z s α t...u t...u = z β t...u j j...j v + T sj β t...u j...j v z s. Substtutng ths at.b t follows that L z t...u = t...u T s j β s z [z α α= t...u z ] + α t...u + Ts α z s t...u ] + α= T α s s α α t...u z β= from where we obtan.8 for λ, µ, ν =,,. From exosed the next theorem s vald: T s j β [z β= s j β t...u z, t...u Theorem. A necessary and suffcent condton for the tensor feld t u to be rgd wth resect to nfntesmal deformatons. s the annulment of the rght sde n any of the equatons.8,.,,, 7, 8. For the examle, based on.7 t follows that a necessary and suffcent condton the tensor feld t u to be rgd s.9 t u ; z = α= z α ; α t u β= z ;j β t u,

10 Infntesmal rgdty and flexblty... 9 what gves a manner for an exresson of covarant dervatve n the drecton of deformaton vector z, that dervatve beng taken by vrtue of symmetrc art L of the connecton L. The Le dervatve and the rgdty of the connecton. On the base of. for the Le dervatve of the connecton we have. L z L = z, + L,z z,l + z,j L k + z,k L j. As t was roved at [MV S] Le dervatve can be wrtten n the next way. L z L = z + R z + Tj,k z + L s T s z + L sk T j s z + Tj z,k,. L z L = L z L z + R z + Tj z k,. L z L = L z L = z + R z + Tj k z + Tk z + T kj z + T z + TsjT k s + TskT j s + TsT z s, j. L z L = L zl z + R z T z + T jz k,.5 L z L = L zl z + R z + Tj k + TsjT k s + TskT jz s + Tkz where [Mn7], [Mn77], [Mn79] j,.6 R = L, L j,k + Ls L s Ls j L sk.7 R = L kj, L j,k + Ls kj L s Ls j L ks.8 R = L, L j,k + L s L s L s jl sk + L s kt sj.9 R = L, L j,k + L s L s L s jl sk + L s kt sj are curvature tensors of the sace L N.

11 L. S. Velmrovć, S. M. Mnčć, M. S. Stankovć.. We have roved at Theorem.. that the Le dervatve of a tensor can be exressed more concse by usng several tyes of covarant dervatves at L N smultaneously. It s the same case for the Le dervatve of the connecton. Namely, the next theorem s n force [VMS ]: Theorem. The Le dervatve of non-symmetrc connecton L s a tensor of the tye, and can be exressed wth resect to covarant dervatves by equatons.. 5, as well as by. L z L = z j k + R z.. L z L = z k j + R kjz... Comarng. and.7, we can see that the Le dervatve of a tensor at the sace L N of symmetrc connecton L and Le dervatve at the sace L N of non-symmetrc connecton L are exressed n the same way: wth resect to gven symmetrc connecton L n the frst case, and n the second wth resect to the symmetrc art L of non-symmetrc connecton L. An analogous roblem can be consdered n the case of a connecton that s not a tensor. At the sace L N of symmetrc connecton L, by reason of T = all the cases of exresses for the Le dervatve consdered before, reduce to. L z L = z ; + R z, where R s curvature tensor, generated by L. Let us examne a sace L N of non-symmetrc affne connecton L, where L, T are gven by.5. The man urose s to exress L z L. by covarant dervatves wth resect to L, and R by R, formed by L. We have [VMS ]:. L z L = L zl z ; + R z + L z T.. L z L = L zl = z ; + R Based on the onted out facts t follows Theorem. Le dervatve of non-symmetrc connecton L can be gven by the equaton., where covarant dervatve denoted by ; and curvature tensor z.

12 Infntesmal rgdty and flexblty... R are formed wth resect to symmetrc art L of the connecton L, and L zt s exressed accordng to.7 wth resect to L. The Le dervatve of symmetrc art of connecton s gven accordng to..e. t s n the same form as for symmetrc connecton equaton.. In relaton wth the rgdty of the connecton, from exosed above t follows Theorem.For the rgdty of a non-symmetrc connecton L wth resect to nfntesmal deformaton., a necessary and suffcent condton s an annulment of the rght sde n any of the equatons. 5,,. All these condtons are equvalent as they sgnfy the annulment of the Le dervatve of the connecton. E.g., from. the cted rgdty condton reduces to z j k = R z. In the case of symmetrc connecton ths condton reduces to z ; = R z, where s taken nto consderaton the skew symmetry of curvature tensor wth resect to the last two ndces. 5 Infntesmal deformaton of curvature tensors At [Mn 7], [Mn 77], [Mn 79] are obtaned at all curvature tensors n L N, and at [Mn 79] s roved that 5 of them are ndeendent. They are.6 9 and 5. R 5 jmn = L jm,n + L mj,n L jn,m L nj,m + L jm L n + L mj L n L jn L m L nj L m. Denotng by semcolon ; covarant dervatve wth resect to symmetrc connecton L, then accordng to [Mn79], we have 5. R jmn = R o jmn + T jm;n T jn;m + T jm T n T jn T m, 5. R jmn = R jmn T jm;n + T jn;m + T jm T n T jn T m, 5. R jmn = R jmn + T jm;n + T jn;m T jm T n + T jn T m T mnt j, 5.5 R jmn = R jmn + T jm;n + T jn;m T jm T n + T jn T m + T mnt j

13 L. S. Velmrovć, S. M. Mnčć, M. S. Stankovć 5.6 R 5 jmn = R jmn + T jm T n + T jn T m. At 5. 6 all the addends at the rght sde are tensors. We wll consder nfntesmal deformatons of cted fve curvature tensors. 5. Infntesmal deformaton of curvature tensor R Accordng to 5. for deformed frst curvature tensor at L N.e. for R, we have 5.7 R jmn x = L jm,n L jn,m + L jm L n L jn L m, and wth resect to L jmx = L jmx + DL jm, one obtans R jmn = L jm + DL jm,n L jn + DL jn,m + L jm + DL jm L n + DL n L jn + DL jn L m + DL m. Develong ths and omttng the members of the form DL DL......, as they nclude ε, we have 5.8 R jmn = L jm,n + DL jm,n L jn,m DL jn,m + L jm L n + L jm DL n + DL jm L n L jn L m L jn DL m DL jn L m. As DL jm s a tensor, we can consder covarant dervatve: wherefrom DL jm n = DL jm,n + L n DL jm L jn DL m L mn DL j 5.9 DL jm,n + L ndl jm = DL jm n + L jn DL m + L mndl j and n the same manner 5.9 DL jn,m + L mdl jn = DL jn m + L jm DL n + L nmdl j If we have n mnd.6, 5.9, 9, the equaton 5.8 becomes 5. R jmn = R jmn + DL jm n DL jn m + Tmn DL j. From here Le dervatve of curvature tensor s 5. L z R jmn = L z L jm n L z L jn + TmnL m z L j.

14 Infntesmal rgdty and flexblty... We can also start from 5., and then we have R jmn = R jmn + T jm;n T jn;m + T jm T n T jn T m =R jmn + DR jmn + T jm + DT jm ;n T jn + DT jn ;m + T jm + DT jm T n + DT n T jn + DT jn T m + DT m. Omttng the members contanng DT DT, we get R jmn = R jmn + DR jmn + T jm;n + DT jm ;n T jn;m DT jn ;m + T jm T n + T jm DT n + DT jm T n T jn T m T jn DT m DT jn T m. So, wth resect to 5., one obtans 5. R jmn = R jmn + DR jmn + DT jm ;n DT jn ;m + DT jm T n T jn T m, where wth ; s denoted covarant dervatve wth resect to L.5, and dvdng wth ε: 5. L z R jmn = L z R jmn + L zt jm ;n L zt jn ;m + L zt jm T n T jn T m. 5. Infntesmal deformaton of curvature tensors R,..., R 5 By analogcal rocedure one obtans the Le dervatve for curvature tensors R,..., R 5. In that manner s 5. L z R = L z L mj L z L jmn n nj m + TnmL z L j. and 5. L z R =L z R jmn jmn + L zt mj ;n L zt nj ;m + L zt jm T n T jn T m,

15 L. S. Velmrovć, S. M. Mnčć, M. S. Stankovć and 5. L z R jmn = L z L jm n L z L nj + T jm L zl n + T jn L zl m + L mnl z T j + L z T jl nm, m and 5.5 L z R L z R = L z R jmn jmn + L zt jm ;n + L zt jn ;m + L zt mt jn T nt jm T jt mn, jmn = L z L jm n L z L nj + T jm L zl n + T jn L zl m + L mnl z T j + L z T jl mn, L z R = L z R jmn jmn m + L zt jm ;n + L zt jn ;m + L zt mt jn T nt jm + T jt mn, 5.6 L z R = 5 jmn [L zl jm L z L n nj + L z L m mj L z L n jn m], and 5.6 L z R 5 jmn = L z R jmn + L zt jm T n + T jn T m. Based on exosed, related to nfntesmal deformatons and rgdty of curvature tensors n L N, the followng theorems are vald: Theorem 5.Le dervatves of the curvature tensors R,..., R 5 n the sace L N of non-symmetrc affne connecton L can be exressed by the equatons 5.,,, 5.6, 6. Theorem 5.If wth resect to nfntesmal deformaton. the connecton s rgd,.e. L z L =, then all curvature tensors R, =,..., 5, are rgd, that s L z R jmn =, wth resect to that deformaton. Conversely, e.g. from L z R jmn =, t follows 5.7 L z L jm n = L z L jn TmnL m z L j. and smlar relatons for the rest curvature tensors. In the case of symmetrc connecton T= the equaton 5.7 and corresondng equatons for the remanng curvature tensors reduce to L z L jm ;n = L z L jn ;m.

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