On Ricci identities for submanifolds in the 2-osculator bundle

Transcript

1 Irnn Journl of Mthemtcl Scences n Informtcs Vol. 8 No pp -2 On Rcc enttes for submnfols n the 2-oscultor bunle On Alexnru Unversty Trnslvn of Brşov eprtment of Mthemtcs n Informtcs Blv. Iulu Mnu no. 50 Brşov Romn. E-ml: lexnru.on@untbv.ro Abstrct. It s the purpose of the present pper to outlne n ntroucton n theory of embengs n the 2-oscultor bunle. Frst we recll the noton of 2-oscultor bunle [9] [2] [4] n the noton of submnfols n the 2-oscultor bunle [9]. A movng frme s constructe. The nuce connectons n the reltve covrnt ervton re scusse n the fourth n ffth secton [5] [6]. The Rcc enttes for the eflecton tensors re presente n the seventh secton. Keywors: nonlner connecton lner connecton nuce lner connecton -torsons n -curvtures Mthemtcs subject clssfcton: 53B05 53B5 53B25 53B40.. Introucton Generlly the geometres of hgher orer efne s the stuy of the ctegory of bunles of jets J k 0 Mπk M s bse on rect pproch of the propertes of objects n morphsms n ths ctegory wthout locl coorntes. But mny mthemtcl moels from Lgrngn Mechncs Theoretcl hyscs n Vrtonl lculus use multvrte Lgrngns of hgher orer ccelertons. From here one cn see the resonof constructon ofthe geometry ofthe totl spce of the bunle of hgher ccelertons or the oscultor bunle of hgher orer n locl coorntes. Receve 2 July 202; Accepte 4 August 203 c 203 Acemc enter for Eucton ulture n Reserch TMU

2 2 On Alexnru As fr we know the generl theory of submnfols n prtculr the Fnsler submnfols [5] s fr from beng settle [] [5] [6] [7]. In [0] n [] R.Mron n M. Anstse gve the theory of subspces n generlze Lgrnge spces. Also n [8] n [9] R. Mronpresente the theoryofsubspces n hgher orer Fnsler n Lgrnge spces respectvely. Ths rtcle s nspre by the orgnl constructon of the hgher orer geometry gven by R. Mron n Gh. Atnsu [9] [2] [3] [4] n the new spects gve by Gh. Atnsu n [] n [2]. If ˇM s n mmerse mnfol n mnfol M nonlner connecton on Osc 2 M nuce nonlner connecton Ň on Osc2 ˇM. We tke the cnoncl N-lner metrc connecton on the mnfol Osc 2 M. Ths llows obtn of the nuce tngent n norml connectons n the ntroucton of the reltve covrnt ervton n the lgebr of -tensor fels [5]. If n [9] R. Mron use the cnoncl metrcl N-lner connecton of the spce L 2n hvng three coefcents Fjk jk 2 jk n ths rtcle we tke the cnoncl metrcl N-lner connecton of the mnfol Osc 2 M hvng nne coefcents L bc bc = 02[5] [6]. 0 bc 2 In ths pper we present the Rcc enttes for the Louvlle -vector fels z n z 2 on the submnfol Osc 2 ˇM. For the Louvlle -vector fels z n z 2 on the mnfol Osc 2 M the problem ws solve by professor Atnsu Gh. n [] n [2]. 2. The 2-oscultor bunle Osc 2 Mπ 2 M Let M be rel fferentble mnfol of menson n. A pont of M wll be enote by x n ts locl coornte system by Uϕϕx = x. The nces b...run over the set {2...n} n Ensten conventon of summrzng s opte ll over ths work. Let us conser two curves ρ : I M hvng mges n omn of locl chrt U M. We sy tht ρ n hve contct of orer 2 n pont x 0 U f: ρ0 = 0 = x 0 0 I n for ny functon f F U : t f ρt t=0= t f t t=0 = 2 2. The relton contct of orer 2 s n equvlence relton on the set of smooth curves n M whch pss through the pont x 0. Let [ρ] x0 be clss of equvlence relton. It wll be clle 2-oscultor spce t the pont x 0 M. The set of 2-oscultor spces t the pont x 0 M wll be enote by Osc 2 x 0 M n we put Osc 2 M = x Osc2 x 0 M 0 M One consers the mppng π 2 : Osc 2 M M efne by π 2 [ρ] x0 = x0. Obvously π 2 s surjecton.

3 On Rcc enttes for submnfols n the 2-oscultor bunle 3 The set Osc 2 M s enowe wth nturl fferentble structure nuce by tht of the mnfol M so tht π 2 s fferentble mpng. It wll be escreb bellow. The curve ρ : I M φimρ U s nlytclly represente n the locl chrtuϕ by x 0 = x 0 = x 0. Tkngthe functon ffrom 2. succesvely equl to the coornte functons x then representtve of the clss [ρ] x0 s gven by x t = x 0+t x t 0+ x 2 t22 0 t I. t2 The prevous polynomls re etermne by the coeffcents x 0 = x 0y = x t 0y2 = 2 x t2 π 2 Hence the pr UΦ wth Φ[ρ] x0 = x 0y y 2 R 3n [ρ] x0 π 2 U s locl chrt on Osc 2 M. Thus fferentble tls A M of the ferentble structure on the mnfol M etermnes fferentble tls A Osc2 M on Osc 2 M n therefore the trple Osc 2 Mπ 2 M s fferentble bunle. We wll entfe the 2-oscultor bunle Osc 2 Mπ 2 M wth 2- tngent bunle T 2 Mπ 2 M. By 2.2 trnsformton of locl coorntes x y y 2 x ỹ ỹ 2 on the mnfol Osc 2 M s gven by x = x x...x n x et x b 0 ỹ = x x byb 2.3 2ỹ 2 = ỹ x b yb +2 ỹ y by2b One cn see tht Osc 2 M s of menson 3n. Let us conser the 2-tngent structure J on Osc 2 M J x = y J = y y 2 J = 0 y 2 where x u y u y 2 u s the nturl bss of the tngent spce T u Osc 2 M u Osc 2 M.If N s nonlner connecton on Osc 2 M then N 0 = NJN 0 = N re two strbutons geometrclly efne on Osc 2 M ll ofmensonn. Let{ usconserthe } strbutonv 2 onosc 2 M locllygenerte by the vector fels. onsequently the tngent bunle of Osc 2 M t y 2 the pont u Osc 2 M s gven by rect sum of the vector spce:

4 4 On Alexnru T u Osc 2 M = N 0 u N u V 2 u u Osc 2 M. 2.4 { } δ δ δ We conser δx δy n pte bss to the ecomposton δy n ts ul bss enote by x δy δy 2 where x = x δy = y +M b x b 2.5 δy 2 = y 2 +M b δy b +M 2 b δy 2b. efnton 2.. A lner connecton on Osc 2 M s clle N-lner connecton f t preserves by prllelsm the horzontl n vertcl strbuton N 0 N n V 2 on Osc 2 M. Any N-lner connecton cn be represente by unque system of functons ΓN = L b b = 02. These functons re clle 0 b 2 the coeffcents of the N-lner connecton. If on the mnfol Osc 2 M N-lner connecton s gven then there exsts n h -v - n v 2 -covrnt ervtves n locl pte bss = 02. Any -tensor T of type rs cn be represente n the pte bss n ts ul bss n the form n we hve T = T...r b...b s δ... 2r x b... δy 2bs T...r b...b s = δ T...r b...b s + L c Tc2...r b 0...b s L r c T...r c c c b 0...b s L b T...r cb b s... L bs T...r cb b s c T...r b...b s = δ T...r b...b s + c Tc2...r b...b s r c T...r c c c b...b s b T...r cb 2...b s... bs T...r cb 2...b s c 2 T...r b...b s = δ 2 T...r b...b s + c Tc2...r b 2...b s r c T...r c c c b 2...b s b T...r cb b s... bs T...r cb b s c δ = δ δy δ 2 = δ δy 2; = 02.

5 On Rcc enttes for submnfols n the 2-oscultor bunle 5 2 Theopertors n reclletheh -v -nv 2 -covrnt ervtves wth respect to ΓN. efnton 2.2. A metrc structure on the mnfol Osc 2 M s symmetrc covrnt tensor fel G of the type 02 whch s non egenerte t ech pont u Osc 2 M n of constnt sgnture on Osc 2 M. Loclly metrc structure looks s follows: where G = g bx x b + g bδy δy b + g bδy 2 δy 2b 0 2 rnk g b = n = 02. efnton 2.3. An N-lner connecton on Osc 2 M enowe wth structure metrc G s s to be N-lner metrc connecton f X G = 0 for every X X Osc 2 M. 3. Submnfols n the 2-oscultor bunle Let M be rel n-mensonl mnfol n let ˇM be rel m-mensonl mnfol mmerse n M v the mmerson : ˇM M. Locly cn be gven n the form x = x u...u m rnk x u = m 3. The nces bc...run over the set {...n} n... run on the set {...m}. We ssume < m < n. If s n embeng then we entfy ˇM wth ˇM n sy tht ˇM s submnfol of the mnfol M. Therefore 3. wll be clle the prmetrc equtons of the submnfol M n the mnfol M. The embeng : ˇM M etermnes n mmerson Osc 2 : Osc 2 ˇM Osc 2 M efne by the covrnt functor Osc 2 : Mn Mn.[9] The mppng Osc 2 : Osc 2 ˇM Osc 2 M hs the prmetrc equtons: x = x u...u m rnk x u = m y = x u v 3.2 2y 2 = y u v +2 y v v2

6 6 On Alexnru where x u = y v y2 = v 2 y u = y2 v. 3.3 The Jcobn mtrx of 3.2 s J Osc 2 n ts rnk s 3m. So Osc 2 s n mmerson. The fferentl of the mppng Osc 2 : Osc 2 ˇM Osc 2 M mps the cotngentspcet Osc 2 M t pont ofosc 2 M ntothe cotngentspce T Osc 2 ˇM t pont of Osc 2 ˇM by the rule: x = x u u y = y u u + y v v 3.4 y 2 = y2 u u + y2 v v + y2 v 2v2. We use the prevous theory to stuy the nuce geometrcl objects from Osc 2 M to Osc 2 ˇM. LetusconserFnslerspce[]F n = MF xy hvngg b xy = 2 F 2 2 y y s the funmentl tensor fel. The restrcton ˇF of the funmentl functon F to the submnfol Osc ˇM s gven b by ˇF uv = F xuy uv n the pr ˇF m = ˇM ˇF s Fnsler spce n t s clle the nuce Fnsler subspces of the Fnsler spce F n. Thereexsts nonlnerconnectonon the mnfol Osc 2 M etermne only by g b xy. The ul coeffcents of ths nonlner connecton re [9] M b = G y b M 2 b = 2 ΓM b M M b where G = 2 γ bc xy y b y c Γ = y x +2y 2 y n γ bc xy re the hrstoffel symbols of the funmentl tensor g b.

7 On Rcc enttes for submnfols n the 2-oscultor bunle 7 Next we conser Bu = x u n G =g b x x b + g b δy δy + g b δy 2 δy 2 the Ssk prolongton { of the metrc } g long Osc 2 M. Thus BB 2...B m re m-lner nepenent -vector fels on Osc 2 ˇM. { } Also B B2...Bn re -covector fels wth respect to the next trnsformtons of coorntes ū = ū u...u m rnk ū u = m v = ū u v v 2 = v u v +2 v. v { } v2 Of course -vector fels B...B m re tngent to the submnfol ˇM. We sy tht -vector fel ξ xy y 2 s norml to Osc 2 ˇM f on ˇπ Ǔ Osc 2 ˇM we hve g b xuy uv v 2 y 2 uv v 2 B u ξ b xuy uv v 2 y 2 uv v 2 = 0. onsequently on ˇπ Ǔ Osc 2 ˇM there exst n m unt vector fels B ā ᾱ =...n m norml long Osc 2 ˇM n to ech other: g b B Bb = 0 g b B ā Bb = δᾱ ᾱ =...n m. 3.6 The system of -vectors B ā ᾱ =...n m s etermne up to orthogonl trnsformtons of the form B ā = A ᾱ Bā Aᾱᾱ On m 3.7 where ᾱ... run over the set 2..n m. We sy tht the system of -vectors {BB } ā etermnes frme n Osc 2 M long to Osc 2 ˇM. Itsulfrmewllbeenoteby { B uv...v 2 Bᾱ uv...v 2}. Ths s lso efne on n open set ˇπ Ǔ Osc 2 ˇM Ǔ beng omn of locl chrt on the submnfol ˇM. The contons of ulty re gven by: Usng 3.6 we euce: B B = δ B Bᾱ = 0 B Bā = 0 Bᾱ Bā = δᾱ 3.8 B B b +Bā Bᾱb = δ b. 3.9 g B = g bb δ ᾱ B b = g bb ā. 3.0

8 8 On Alexnru So we cn look t the set { R = uv v 2 ;BuB uv ā v 2} uv v 2 ˇπ Ǔ s movng frme. Now we shll represent n R the -tensor fels from the spce Osc 2 M restrcte to the open set ˇπ Ǔ. 4. Inuce nonlner connectons Now let us conser the cnoncl nonlner connecton N on the Osc 2 M. Then ts ul coeffcents M b M 2 b epens only by the metrc g. We wll prove tht the restrcton of the nonlner connecton N to Osc 2 ˇM unquely etermnes n nuce nonlner connecton Ň on Osc2 ˇM. Of course Ň s well etermne by mens of ts ul coeffcents ˇM ˇM or by mens of ts 2 pte cobss u δv δv 2. efnton 4.. A non-lner connecton Ň on Osc2 ˇM s clle nuce by the nonlner connecton N f we hve δv = B δy δv 2 = B δy roposton 4.2. [6] The ul coeffcents of the non-lner connecton Ň re ˇM = B ˇM = 2 B B0 +M b Bb Bδγ 2 u vδ v γ +B δ v2δ +M b B b 0 +M 2 b Bb where M b M 2 b re the ul coeffcents of the non-lner connecton N. 4.2 Theorem 4.3. [6] The cobss x δy δy 2 restrcte to Osc 2 ˇM s unquely represente n the movng frme R n the followng form: x = B u δy = B δv +B ā K ᾱ u 4.3 δy 2 = B δv2 +B ā K δv +B ā K 2 u

9 On Rcc enttes for submnfols n the 2-oscultor bunle 9 where ᾱ K = Bᾱ B0 +M b Bb ᾱ K = 2 Bᾱ Bᾱf Bγ Bδγ 2 u vδ v γ +B b δ v2δ +M b B b 0 +M 2 Bγ f f +M b Bb γ B0 +M gb g b Bb 4.4 re mxe -tensor fels. roof. The frst relton s obvous. From 3.2 n 4.2 we obtn 4.3. Generlly set of functons T...ᾱ j... uv v 2 whch re -tensors n the nex j... n -tensors n the nex...n tensors wth respect to the trnsformtons 3.7 n the nex ᾱ... s cll mxe -tensor fel on Osc 2 ˇM. 5. The reltve covrnt ervtves We shll construct the opertors of reltve or mxe covrnt ervton n the lgebr of mxe -tensor fels. It s cler tht wll be known f ts cton on functons n on the vector fels of the form X xuy uv y 2 uv v 2 X uv v 2 Xᾱ uv v 2 5. re known. LetbethecnonclN-lnermetrcconnectononthemnfolOsc 2 M [2] L bc = 2 g δ b g c +δ c g b δ g bc 00 L bc 0 = B bc + jj 2 g f δ c g b B cb g f f B c g bf j = 2 jj jj bc = 2 g δ b g b k = 02 k 5.2 bc = 2 g 2c g b l = 0 l2 bc = 2 g δ b g c +δ c g b δ g bc = 2.

10 0 On Alexnru efnton 5.. The couplng of the cnoncl N-lner metrc connecton wth the nuce nonlner connecton Ň long Osc2 ˇM s loclly gven by the set of ts nne coeffcents ĎΓ Ň = Ľ bδ Č bδ Č bδ = 02 where 2 0 Ľ bδ = L b B δ + b B δk + δδ bδ B δk 2 δδ Č bδ = b B δ + b B δk 2 δδ = Č bδ = 2 b 2 B δ. We hve the opertors Ď n = 02 wth the property Ď X = X moulo where n X = X +X b ω b 5.5 Ď X = X +X b ˇω b. 5.6 Here ω b n ˇω b re the -forms of the cnoncl N-lner metrc connecton n of the couplng Ď respectvely. Of course we cn wrte Ď X n the form Ď X = X δ u δ +X δ δv δ +X 2 δ δv 2δ. efnton 5.2. We cll the nuce tngent connecton on Osc 2 ˇM by the cnoncl N-lner metrc connecton the set of ts nne coeffcents Γ Ň = L δ δ = 02 where 0 δ 2 L δ = 0 B B δ +Bf Ľ fδ 0 δ = B Bf Č fδ = δ = B Bf Č 2 fδ. 2

11 On Rcc enttes for submnfols n the 2-oscultor bunle We hve the opertors wth the propertes X = Bb Ď Xb for X = BγX γ 5.8 X = X +X ω 5.9 where ω re the connecton -forms of = 02. As n the cse of Ď we my wrte X = X δ uδ +X δ δv δ +X 2 δ δv 2δ. efnton 5.3. We cll the nuce norml connecton on Osc 2 ˇM by the cnoncl N-lner metrc connecton the set of ts nne coeffcents Γ Ň = ᾱ δ ᾱ δ where L 0 2 ᾱ δ δb L 0 ᾱ δ = Bᾱ δu δ +Bf Ľ fδ 0 δb ᾱ δ = Bᾱ δv δ +Bf Č fδ B 2 ᾱ δ = Bᾱ v 2δ +Bf Č fδ 2 As before we hve the opertors wth the propertes. = Xᾱ = Bᾱb Ď Xb for X = B γx γ ā 5. Xᾱ = Xᾱ +X ω ᾱ 5.2 where ω ᾱ re the connecton -forms of = 02. We my set Xᾱ = Xᾱ δ uδ +Xᾱ δ δv δ 2 +Xᾱ δ δv 2δ. Now we cn efne the reltve or mxe covrnt ervtves enounce t the begnng of ths secton.

12 2 On Alexnru Theorem 5.4. A reltve mxe covrnt ervton n the lgebr of mxe -tensor fels s n opertor for whch the followng propertes hol: f = f f F Osc 2 ˇM X = Ď X X = X Xᾱ = Xᾱ = 02. The connecton -forms ˇω b ω ω ᾱ wll be clle the connecton -forms of. The Louvlle vector fels for submnfols ntrouce by professor Mron n [9] re γ = v v 2 2 γ = v +2v2 v v 2. If we represent ths vector fels n the pte bss we get γ = z 2 2 γ = z δ +2z 2 2 where z = v z 2 = v M v. -vector fels z n z 2 re clle the Louvlle -vector fels of submnfol Osc 2 ˇM. The z - n z 2 -eflecton tensor fels re: z = z = 2 z = 2 z 2 = 2 2 z = z =. 5.3 roposton 5.5. The z -eflecton fels hve the expresson: = N +z γ L 0 γ = δ +zγ γ = z γ 2 γ.

13 On Rcc enttes for submnfols n the 2-oscultor bunle 3 roof. Inee f we tke z = δ z +z γ L 0 γ j z = δ j z +z γ γ = 02;j = 2;δ 2 = 2 j we fn the Formule 5.4. roposton 5.6. The z 2 -eflecton fels re gven by 2 = N +M zγ δ N γ +z 2γ L γ 0 2 = 2N N zγ B γ +z 2γ γ 5.5 = δ + 2 zγ B 2 γ +z 2γ 2 γ. roof. Inee f we tke z 2 = δ z 2 +z 2γ L 0 γ j 2 z = δ j z 2 +z 2γ γ = 02;j = 2;δ 2 = 2 j we fn the Formule Apte components of torson n curvture tensors The stuy of the pte components of the torson n curvture tensors of n rbtrry N-lner connecton ΓN on Osc 2 M ws one n [2] n []. In wht follows we stuy the pte components of the torson n curvture tensors for the reltve or mxe covrnt ervtves = 02. Theorem 6.. In locl coorntes the torson -tensors of the reltve or mxe covrnt ervtves hve the next expresons: T 00 = L L 00 γ 00 T 0 = R = δ γn δ N γ 0 T 02 = R = δ γn δ N γ N δ γ N δ N γ

14 4 On Alexnru = = 0 20 = 2γ N 2 = δ γn L 0 γ = δ γn δ N γ +N δ γ N 2 = 2γ N +N 2γ N L 2 γ S = Q = 2 S = γ S = R = δ γn δ N γ 2 2 Q = 2γ N S = 2 γ γ. roof. Usng the generl locl expressons from [2] n [] whch gve the - components of the torson tensor of n N-lner connecton ΓN we euce tht the pte components of the mxe covrnt ervtves = 02 re gven by the formuls from theorem. The followng -tensor fels wll be neee n our clcultons. T 0 = L L 0 γ 0 jj = B jj L 0 γ Q = B 2 γ S j = j j γ 6.2 = 02;j = 2. We remrk tht we hve 0 T 0 = T 00 j jj = jj 2 Q = 2 Q j S j = S j j = 2.

15 On Rcc enttes for submnfols n the 2-oscultor bunle 5 Theorem 6.2. In locl coorntes the curvture -tensors of the reltve or mxe covrnt ervtves hve the next expresons: R 0 b γδ = δ δ L 0 bγ δ L 0 bδ + L 0 e bγ L 0 eδ L 0 e bδ L 0 eγ + + b R 0 γδ + 2 b R 02 γδ b γδ 2 b γδ = δ δ L 0 bγ bδ γ + = 2δ L 0 bγ 2 bδ γ + b b 2 γδ + γδ + b γδ 2 2 b γδ 2 Q b γδ 2 = 2δ bγ δ γ 2 bδ + e bγ 2 eδ e bδ eγ b 2 γδ S b γδ = δ δ bγ δ γ bδ + e bγ eδ e bδ eγ b R γδ 2 2 S 2 b γδ = 2δ 2 bγ 2γ bδ + 2 e bγ 2 eδ 2 e bδ 2 eγ n R 0 γδ = δ δ L 0 δ L 0 δ + L 0 L 0 δ L 0 δ L 0 γ + + R 0 γδ + 2 R 02 γδ γδ = δ γ L 0 δ γ + γδ + γδ γδ = 2δ L 0 2 δ γ + 2 γδ + γδ 2

16 6 On Alexnru Q γδ 2 = 2δ δ γ 2 δ + 2 δ δ γ γδ S γδ = δ δ δ γ δ + δ δ γ R γδ 2 2 S 2 γδ = 2δ 2 2γ δ δ 2 δ 2 γ n R 0 γδ = δ δ L 0 δ L 0 δ + L L 0 0 δ L L 0 δ γ R γδ + 0 R 2 γδ 02 γδ = δ γ L 0 δ γ + γδ + 2 γδ 2 2 γδ = 2δ L 0 2 δ γ + 2 γδ + 2 γδ Q 2 γδ = 2δ δ γ + 2 δ 2 δ δ γ γδ 2 S γδ = δ δ δ γ + δ δ δ γ R 2 γδ 2 S 2 γδ = 02; 2 = 2δ 2 2γ + δ 2 = 2 2 δ 2 δ γ 2 ; = ; R = 0;δ 0 = δ δ 2 =

17 On Rcc enttes for submnfols n the 2-oscultor bunle 7 roof. The generl formuls tht express the locl curvture -tensors of n rbtrry N-lner connecton for more etls see [2] n [] pple to the reltve covrnt ervtves = 02 mply the bove formuls. 7. The Rcc enttes Let ĎΓ Ň = Ľ bδ Č bδ Č bδ be the couplng of the cnoncl N- 0 2 lner metrc connecton 5.2 wth the nuce nonlner connecton N long the mnfol Osc 2 M Γ Ň = L δ δ δ n Γ Ň = 0 2 ᾱ δ ᾱ δ = 02 the nuce tngent connecton on Osc 2 ˇM L 0 2 ᾱ δ n the nuce norml connecton on Osc 2 ˇM respectvely. Theorem 7.. [3] For ny -vector fels X the followng Rcc enttes hol: X γ X γ = X δ R δ T 0 0 X R 0 X R X 2 02 X γ X γ = X δ δ X X 2 2 X X 2 γ X 2 γ = X δ δ X X X 2 7.

18 8 On Alexnru X 2 γ X 2 γ = X δ Q δ X 2 2 Q X j j γ X j γ j = X δ S where R = 0 = 02j = 2 n X 2 δ S j j R X 2 j2 X = X 0 δ +X δ +X 2 2 s n rbtrry -vector fel on the submnfol Ě = Osc2 ˇM. X roof. Let Y A n ω A where A { = 02} be on Ě = Osc2 ˇM the bses n the ul bses pte to the nonlner connecton N n let X = X F Y F be -vector fel on E. In ths context usng the followng true equltes pple for the nuce tngent connecton Γ Ň : Y Y B = Γ F B Y F 2 [Y B Y ] = R F B Y F 3 TY Y B = T F B Y F = {Γ F B ΓF B RF B }Y F 4 RY Y B Y A = R F AB Y F 5 Y ω B = Γ B F ωf 6 [RY Y B X] ω B ω = { Y YB X YB Y X [YY B]X} ω B ω by rect clculton we fn tht X A :B: XA ::B = XF R A FB XA :F TF B 7.2 where :G represents one from the locl covrnt ervtves δ δ or 2 δ prouce by the nuce tngent connecton Γ Ň. Tkng nto ccount n 7.2 tht the nces AB... belong to the set { = 02} by complcte computtons we fn wht we were lookng for. The Rcc enttes 7. pple to the Louvlle -vector fels z n z 2 le to the next theorem.

19 On Rcc enttes for submnfols n the 2-oscultor bunle 9 Theorem 7.2. The eflecton tensor fels stsfy the followng enttes: j γ j γ = z jδ R 0 δ j δ δ T 0 j δ δ R j2 δ 0 δ R 02 j γ j γ j 2 γ j2 γ = z jδ δ j δ δ j δ j2 δ δ 2 = z jδ δ j 2 δ 2 j δ 2 δ j2 δ 7.3 j 2 γ j2 γ = z jδ Q δ X 2 j δ 2 δ j2 j l γ j γ l = z jδ S l δ δ δ Q = 02;jl = 2; R = 0. jl l δ δ S j2 δ δ R l2 Also f the z -n z 2 -eflecton tensors hve the followng prtculr form = 0 = δ 2 = 0 2 = 0 2 = 0 = δ 7.4 then the funmentl enttes from 7.3 re very mportnt especlly for pplctons.

20 20 On Alexnru roposton 7.3. Wth the eflecton tensor whch re gven by 7.4 the followng enttes hol: z jδ R 0 δ = R 0j z δ 2 δ = 2 z 2δ δ = 2 z jδ j δ = jj z δ Q δ = 2 2 z 2δ Q δ = Q 2 z jδ S j δ = j S = 02;j = 2. z δ S 2 δ = 0 z 2δ S δ = R roof. Usng the Rcc enttes of the Louvlle -vector fels z n z 2 fromthe lsttheoremnthe prtculrformofthe z -n z 2 -eflecton tensors from 7.4 we get the Formule 7.5. Remrk 7.4. The eflecton -tensor enttes 7.3 wll be use n the ner future for the constructon of the geometrcl Mxwell equtons tht wll govern the bstrct electromgnetsm n the Lgrnge subspces of secon orer ths s our work n progress. 8. Acknowlegments Ths work ws supporte by ontrct wth Snoptx No. 844/2009. The uthor woul lke to express hs grttue to the referees for creful reng n helpful comments. References [] Gh. Atnsu New Aspects n erentl Geometry of Secon Orer Trtu Unv. ress [2] Gh. Atnsu New spects n fferentl geometry of the secon orer Semnrul e Mecncă Ssteme nmce ferenţle Unv. e Vest n Tmşor Fcultte e Mtemtcă vol pp. -8. [3] Gh. Atnsu A. On The Rcc enttes on submnfols n the 2-oscultor bunle roc. of the 4th Ntonl Semnr on Fnsler Lgrnge n Hmlton Spces Brşov [4] Gh. Atnsu Structure equtons of secon orer BSG roccengs 3 Geometry Blkn ress Buchrest Romn [5] A Bejncu H.R. Frn The Geometry of seuo-fnsler Submnfols Kluwer Ac. ubl [6] N. Broojern E. eyghnb n A. Heyrc fferentton long Multvector Fels Irnn Journl of Mthemtcl Scences n Informtcs [7] S. Hong M.M. Trpth Rcc curvture of submnfols of Sskn spce form Irnn Journl of Mthemtcl Scences n Informtcs [8] R. Mron The Geometry of Hgher Orer Fnsler Spces Hronc ress USA 998.

21 On Rcc enttes for submnfols n the 2-oscultor bunle 2 [9] R. Mron The Geometry of hgher orer Lgrnge spces. Applctons to mechncs n physcs Kluwer Ac. ubl. FTH no [0] R. Mron M. Anstse Vector bunles. Lgrnge Spces. Applctons to Reltvty E. Aceme Române 987. [] Mron R. Anstse M. The Geometry of Lgrnge Spces: Theory n Applctons Kluwer Ac. ubl. FTH no [2] R. Mron Gh. Atnsu Hgher orer Lgrnge spces Rev. Roum. e Mth. ures et Appl. T [3] R. Mron Gh. Atnsu fferentl Geometry of the k-oscultor Bunle Rev. Roumne Mth ures et Appl. T [4] R. Mron Gh. Atnsu Lgrnge Geometry of Secon Orer Mth. omput. Moellng [5] A. On The Guss-Wengrten formule of secon orer BSG roceengs pp [6] A. On The reltve covrnt ervte n nuce connectons n the theory of embengs n the 2-oscultor bunle Bull. Trnslvn Unv