M-LINEAR CONNECTION ON THE SECOND ORDER REONOM BUNDLE

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1 STUDIA UNIV. AEŞ OLYAI, MATHEMATICA, Volume XLVI, Number 3, September 001 M-LINEAR CONNECTION ON THE SECOND ORDER REONOM UNDLE VASILE LAZAR Abstract. The T M R bundle represents the total space of a tme dependent geometry of second order. In ths bundle t s studed a specal class of dervaton rules, named M-lnear connectons.. There are gven ther characterzaton and t s proved ther exstence. Fnally there are studed geometrcal propertes of one M-lnear connecton. 1. Introducton The study of the tme dependent Lagrange geometry (geometry of the reonom Lagrange spaces ) was mposed from consderatons of mechanc,a systematcally study of ths s fndng n the M.Anastase and H.Kawaguch paper [1],[],[3]. On the other hand, research from the last years mposed nto attenton the consderatons n the superor order geometres where the total space s the prolongaton of k order of the T M tangent bundle of a dfferental manfold or an assocated bundle named the osculator bundle of k order ( [5],[8],[13] ). From calculaton reasons we wll approach here the case k =. The study of the second order reonom bundle E = T M R was done by us n a prevous work([6],[7]). Let M be a dfferentable manfold, dmm = n, x = (x ) the local coordnates n a map (U, ϕ). We are consderng T M the -jets bundle to the tangent curves n x M. Locally on T M the coordnates are u = (x, y, z ) wth the followng rule of change on the ntersecton of two local maps: x = x (x j ) (1.1) ỹ = x x j y 1991 Mathematcs Subject Classfcaton. 53C60, 53C05. Key words and phrases. connectons, reonom bundle. 67

2 one. T M VASILE LAZAR z = 1 ỹ x k yk + x x k zk has a structure of fbre bundle over R n space,whch s not vectoral The reonom bundle of second order s the bundle of drect product E = T M R, n whch varable on R s denoted by t and t s consdered n applcatons as beng the tme. In respect to the (1.1) changes on E we wll haw also and t = t. Takng as a base the E manfold, we wll develop a geometrcal technques of dervaton the sectons on T E. The tangent space T u E present approachng dffcultes due to the fact that the natural bases { x, y, z, } t s changng wth the two order dervatves of x x j. In order to elmnate ths nconvenent we wll consder an adapted base of a nonlnear connecton on E. Let Π : E M the canoncal projecton and Π the cotangent map,v E = KerΠ the vertcal subbundle of second order. We are consderng also the bundle Π 1 : E T M R and VE = KerΠ 1 the vertcal subbundle of frst order, that at hs turn, s subbundle of the vertcal bundle of second order, through hs natural structure. Local bases n VE and V E are respectvely { x, t } and { y, z, }. Defnton 1. A nonlnear connecton on E s a splttng of the T E n the sum T E = V E N E,where N E wll be named the normal subbundle of E. Locally, a base n u N u E dstrbuton s gven by { x = x N j y j M j z j K0 I } We are mposng further the condtons of global defnton of the t adapted felds { y } and { x }, x = xj x x j and y = xj x ỹ j (1.) Consequently, we are obtanng the next changng rules of the nonlnear connecton coeffcents on E. 68 Ñ r x r k x k = xr x k N k x r x x k zk + M r x k k x x r x x k y 1 3 x r x x j x k y y k.. (1.3) = xr x k Mk x r x x k yk (1.4)

3 M-LINEAR CONNECTION ON THE SECOND ORDER REONOM UNDLE K 0 x k x = K0 (1.5) and analogue wth (1.3) and (1.5) for H j and H 0.In consequence we wll take H j = Mj and H0 = K0 n the followng. Gvng { a nonlnear connecton on E s obtanng the next adapted local base for T u E : x, y, z, } that s changng as the vectors as t results from(1.) f there are verfed the condtons (1.3),(1.4),(1.5). Consderng a nonlnear connecton fxed on E, we name d-tensor of (r, s) type a real functon t 1...r j 1...j s (x, y, z, t) that s changng after rule: t h1...hr k 1...k s (ũ) = xh1 x 1... xhr xj1 xjs.... r k1 x x x ks t1... r j 1...j s (u). (1.6) On E we can ntroduce relatvely to the gven nonlnear connecton, the followng geometrcal structures. and hs dual F j = dx y j + y z j + t (1.7) Fj = y x j + z y j + t. ) (1.7 The trplet (F,, t) verfes the condtons : F 3 = t, t( ) = 1 and rank F = n + 1 and t s named the cotangent structure of second order ([1]) Structure ϕ = F F 3 t s an almost tangent of second order structure on E ([1]), rank ϕ = n. adjont to F. The trplet (F,, t) t s also a cotangent structure of second order named Analogue ϕ = F F 3 t s a tangent structure of second order adjont to ϕ.easly there can be deduced lnks between these structures ([6])... Lnear d-connectons on E Let E = T M R be the reonom bundle of second order endowed wth a nonlnear connecton convenently chosen NΓ = (M j, N j, K j ) that determnes the T E = VE HE N E decomposton, wth the correspondng projectors.a feld X X (E) wll be decomposed n X = vx + hx + nx. 69

4 VASILE LAZAR Defnton. It s named lnear d-connecton on E a D lnear connecton on E that preserves trough parallelsm the dstrbutons VE, HE, N E. Theorem 1. A lnear connecton D on E s a d-connecton f and only f there are verfed one of the followng condtons : a) (v + h)d X ny = 0, (v + n)d X hy = 0, (h + n)d X vy = 0 b) D X Y = vd X vy + hd X hy + nd X ny c) Dv = Dh = Dn = 0 d) DP 1 = 0, DP = 0 DP 3 = 0 where P 1 = (n+h) v,p = (n+v) h, P 3 = (v + h) n there are almost product structure on E. The proof results from the fact that: D X ny N E, D X hy HE, D X vy VE. ecause D s a R-lnear applcaton that can be extended to the whole d-tensors algebra, t results that : Proposton. It s only one operator of covarant dervaton DX n named normal dervaton thus that : DXY n = D nx Y and DXf n = (nx)f : X, Y X (E), f F(E). (.1) Locally D n can be expressed the followng way : D n x k D n x j x k = L 1 x D n x k y j = L y (.) y j = L 3 z + L0 ; D n = L0 0k z + L0 ok x k Analogous t s defned the D h covarant h dervaton wth the followng local expressons. 70 D h y k x j = F 1 D h y j = F y k x ; D h z k z j = 3 F z + F 0 y ; D h = F 0k z + F 0 0k y k (.3)

5 M-LINEAR CONNECTION ON THE SECOND ORDER REONOM UNDLE and n totally the same way t s ntroduced D v covarant v dervaton wth local expressons D v. x j z k D v. y j z k = C 1 x = C y ; D v. z k ; D v = C0k z + C0 0k = C00 0 (.4) D v. z j = C 3 z z k The curvatures and torsons of a lnear d connecton are wrtten and are fndng ther local expressons through the drect calculaton.([6]) 3. M-lnear connecton on E Let D be a lnear d connecton on E wth local coeffcents gven by (.1);(.);(.3). Defnton 3. A d lnear connecton D on E t s sad that t s a M lnear connecton (Mron -connecton) f: 1 L= L = L 3 ; 1 = F = F 3 ; F 1 C= C = C 3 (3.1) Let F and ϕ the almost cotangent structures of second order and respectvely second order tangent locally gven by (1.7) and ϕ = F F 3,and (F, ϕ ) ther adjont structures: Defnton 4. a) A D- lnear connecton on E s a F -lnear connecton( respectvely F ) f D = 0 and D = 0 (respectvely DF = 0, D = 0). b) A D lnear connecton on E s a (ϕ, ϕ ) -lnear connecton on E f DF = DF = 0 and D = 0 c) A D lnear connecton on E s a ϕ lnear connecton (respectvely ϕ lnear connecton ) f Dϕ = 0 (respectvely Dϕ = 0 ) d) A D lnear connecton on E s a (ϕ, ϕ ) lnear connecton f Dϕ = Dϕ = 0 Proposton 3. A D -lnear connecton on E s a (F, F ) -lnear connecton f and only f s a (ϕ, ϕ ) -lnear connecton. 71

6 VASILE LAZAR Proof. From DF = 0 DF 3 = 0 D(F F 3 ) = 0 Dϕ = 0 and from DF = 0 D(F F ) = 0 Dϕ = 0. Recprocal, we have ϕ F 3 = 0 and F 3 ϕ = 0 (takng nto account that F 3 (X u ) V u E).It results that DF 3 = 0 and together wth Dϕ = 0, Dϕ = 0 we are obtanng that D(ϕ + F 3 ) = DF = 0 and (Dϕ + F 3 ) = DF = 0. E (F, F ). Proposton 4. A (F.F )-lnear connecton s a d lnear connecton on Proof:Is a (F, F ) lnear connecton s a (ϕ, ϕ ) lnear connecton and usng the fact that v = ϕ ϕ, ϕ = n and ϕ ϕ ϕ ϕ = h t results that Dn = Dh = Dh = 0 s a d lnear connecton on E. Theorem 5. A D lnear connecton on E s a M lnear connecton f and only f s a (F, F )-lnear connecton. Proof:From the proposton 3. t results that f D s a (F, F )-lnear connecton than t s also a d-lnear connecton. ecause (D F )( x j ) = (Dn F )( x j ) = (Dn F )( x k x k x k = D n x k y j L 3 F ( x ) = ( L 3 L x j ) F Dn x k ) y. x j = We are obtanng that (D )( x j ) = 0 L= L 3.In an analogue way, takng these x k values of the adapted base felds, yelds that DF = DF = 0,and hence D s a M lnear connecton on E. gven by : We are wakng the notfcatons F 3 = p and q = I p. Theorem 6. There exsts M lnear connectons on E. One of them s DX Y = DqX qy + DqX py + DpX qy + DpX py (3.) where: [( DqX qy = ϕ v + h ) ] [( X, ϕ y + v n + h ) ] X, vy + 7

7 M-LINEAR CONNECTION ON THE SECOND ORDER REONOM UNDLE [( ϕn n + h ) ] [( X, ϕ hx + ϕ v n + h ) ] [( X, ϕhy + ϕ n + h ) ] X, ny DqX py = p [qx, py ] (3.3) DpX qy = 1 [ {ϕ px, ϕ Y ] + ϕ [ px, ϕ Y ] + ( h [ + n) px, (v + h ]}+ )Y {ϕn [px, ϕ hy ] + ϕ v [px, hy ].} DpX py = 0 px py t(x)t(y ) 0 (3.4) and 0 s a lnear connecton on E. Proof. Trough the drect calculaton t s verfed that D s a lnear connecton and that Dϕ = Dϕ = 0, so D s a M lnear connecton. Gven to X and Y values of the adapted base, from(3.3) results : Corollary 7. The followng functons on E L k j = Ml j z Mk l + N j k z ; Fj k = Mk j z ; Cj k = 0 L k 0 = L k 0j = F k 0j = C 0 0 = C 0 0j = C 0 00 = 0. (3.5) defnng the coeffcents of a M lnear connecton on E,named erwald connecton n the reonom bundle of second order An nterestng problem s the determnaton of the M lnear connecton compatble wth respect to a gven metrc structure on E. We wll approach ths n a comng paper. References [1] M. Anastase and H. Kawaguch: A geometrcal theory of tme dependent Lagrangans.I. Nonlnear connectons. Tensor, N.S., 48(1989), [] M. Anastase and H. Kawaguch: A geometrcal theory of tme dependent Lagrangans.II M- connectons, Tensor, N.S., 48(1989), [3] M. Anastase and H. Kawaguch: A geometrcal theory of tme dependent Lagrangans.III. Applcatons, Tensor, N.S., 49 (1990). [4] Gh. Atanasu and Gh. Munteanu: New aspects n geometry of tme dependent generalzed metrcs, Tensor, N.S., 50(1991), [5] R. awmman: Second order connectons, Journal of Dfferental Geometry 7(197), [6] V.Lazăr: Doctoral thess. [7] V. Lazăr: Nonlnear connecton on total space of second order reonomc geometry, Proc. of Nat. Sem on Fnsler, Lagrange and Hamlton spaces, raşov, 199 (to appear). [8] R. Mron: Hger order Lagrange geomety, Kluwer Acad.Publ.Co.Dordrect

8 VASILE LAZAR [9] R. Mron and M. Anastase: The geometry of Lagrange spaces. Theory and applcaton, Kluwer, Dordrecht, [10] R. Mron and Gh. Atanasu: Compendum sur les espaces Lagrange d, ordre supereur Unv. Tmşoara, Semnarul de matematcă 40 (1994). [11] Gh. Munteanu and Gh. Atanasu: On Mon-connectons n Lagrange spaces of second order, Tensor, N.S., 50(1991), [1] Gh. Munteanu and V. Lazăr: On almost second order almost cotangent structures, Tensor, N.S., 50(1993), [13] K. Yano and S. Ishhara: Tangent and cotangent bundles, M. Dekker Inc Department of Geometry, Unv. Translvana, Str. I. Manu, 50, 00 raşov, Romana 74