Monica Purcaru and Nicoleta Aldea. Abstract

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1 FILOMAT (Nš) 16 (22), 7 17 GENERAL CONFORMAL ALMOST SYMPLECTIC N-LINEAR CONNECTIONS IN THE BUNDLE OF ACCELERATIONS Monca Purcaru and Ncoeta Adea Abstract The am of ths paper 1 s to fnd the transformaton for the coeffcents of an N-near connecton on E = Osc 2 M, by a transformaton of nonnear connectons, to defne n the bunde of acceeratons the genera conforma amost sympectc N-near connecton noton and to determne the set of a genera conforma amost sympectc N-near connectons on E. We treat aso some speca cases of genera conforma amost sympectc N-near connectons on E. 1 Introducton The terature on the hgher order Lagrange spaces geometry hghghts the teoretca and practca mportance of these spaces: [4] [7]. Motvated by concrete probems n varatona cacuaton, hgher order Lagrange geometry, based on the k-oscuator bunde noton, has wtnessed a wde acknowedgment due to the papers: [4] [7], pubshed by Radu Mron and Gheorghe Atanasu. The geometry of k-oscuator spaces presents not ony a speca theoretca nterest, but aso an appcatve one. Due to ts content, the present paper contnues a trend of nterest wth a ong tradton n the modern dfferenta geometry,.e. the study of remarkabe geometrca structures. In the present paper we fnd the transformatons for the coeffcents of an N-near connecton on E = Osc 2 M, by a transformaton of nonnear connectons, ( 2). We defne the genera conforma amost sympectc N-near connecton noton on E, ( 3). 1 2 Mathematcs Subject Cassfcaton. 53C5. Key words and phrases. 2-oscuator bunde, genera conforma amost sympectc N- near connecton, conforma amost sympectc d-structure 7

2 8 Monca Purcaru and Ncoeta Adea We determne the set of a these connectons and we treat aso some speca casess of genera conforma amost sympectc N-near connectons on E, ( 4). Ths paper s a generazaton of the papers:[1] [13]. Concernng the termnoogy and notatons, we use those from: [4], [9], whch are essentay based on M.Matsumoto s book: [2]. 2 The set of the transformatons of N-near connectons n the 2-oscuator bunde Let M be a rea n-dmensona C -manfod and et (Osc 2 M, π, M) be ts 2-oscuator bunde, wth E = Osc 2 M the tota space. The oca coordnates on E are denoted by: (x, y, y (2) ), brefy: (x, y, y (2) ). If N s a nonnear connecton on E, wth the coeffcents (N j, N (2) j ), then et D be an N-near connecton on E, wth the coeffcents DΓ(N) = = (L jk, C jk, C (2)jk ). If N s another nonnear connecton on E, wth the coeffcents N j(x, y, y (2) ), N (2) j(x, y, y (2) ), then there exst a unquey determned tensor feds A j τ 1 1(E), (α = 1, 2) such that: (2.1) N j = N j A j, (α = 1, 2). Conversey,f N j and A j, (α = 1, 2) are gven, then N j, (α = 1, 2), gven by (2.1) s a nonnear connecton. Let us suppose that the mappng N N s gven by (2.1). Accordng to Cap.III, 3.3, [4], we have: D x k y j = L jk, D y y (β)k (β = 1, 2; α =, 1, 2; y () = x ) and = N j x x N j y j (2) y (2). y j = C y (2)j, (β)jk, y = N j y y, y (2)j y (2) = It foows frst of a that the transformaton (2.1) preserve the coeffcents C (2)jk.

3 Genera Conforma Amost Sympectc N-Lnear Connectons 9 Takng n account the fact that: x = t foows: D x k y (2)j = D x k x + A = D x k = L jk + A y (2) D y k j y (2)j + A y j + A y (2)j k C j = L jk j (2), y (2)j y (2) = D ( k D ( +N y m y (2)m ) y (2) + A = (L jk + A k C j + A =D y k y (2)j = D y k y (2)j + A = C y (2)j k D y (2) = y x + A j, y (2)j x k +A +A k y (2) k y (2) ) = y (2)j y (2)j + A k N m C (2)jm k N m C jk = (Cjk + A (2)jm + A y (2) = D ( y y (2)j = C jk (2) k D = y (2) y (2)j y (2) + A (2) k C (2)j = y (2) (2) k C (2)j ). y (2) +A k k y (2) ) = y (2)j y (2) + A k C (2)j ). y (2) Therefore the change we are ookng for s: L jk = L jk + A k C j + A (2.2) C jk = Cjk + A C(2)jk = C So, we have proved: (2)jk. k C (2)j, k N m C (2)jm + A k C (2)j = y (2) (2) k C (2)j Proposton 2.1 The transformaton (2.1) of nonnear connectons mpy the transformatons (2.2) for the coeffcents DΓ(N) = (L jk, C jk, C (2)jk ) of the N-near connecton D. Now, we can prove: Theorem 2.1 Let N and N be two nonnear connectons on E, wth the coeffcents (N j, N (2) j ), (N j, N (2) j)-respectvey. If DΓ(N) = (L jk, C jk, C (2)jk ) and DΓ(N) = (L jk, Cjk, C(2)jk) are two N-, respectvey N-near connectons on the dfferentabe manfod E, then there exsts ony one quntet of tensor feds (A j, A (2) j, B jk, D jk, D (2)jk ) such that:

4 1 Monca Purcaru and Ncoeta Adea (2.3) N j = N j A L jk = L jk + A j k C Cjk = Cjk + A, (α = 1, 2), j + A k C k N m C (2)j D jk, (2)jm + A (2) k C (2)j B jk, C(2)jk = C(2)jk D (2)jk. Proof. The frst equaty (2.3) determnes unquey the tensor feds A j, (α = 1, 2), [3]. Snce Cjk, (α = 1, 2) are tensor feds, the second equaton (2.3) determnes unquey the tensor fed Bjk. Smary the thrd and the fourth equaton (2.3) determne the tensor feds Djk and D (2)jk respectvey. We have mmedatey: Theorem 2.2 If DΓ(N) = (L jk, C jk, C (2)jk) are the coeffcents of an N- near connecton D on E and (A j, A (2) j, B jk, D jk, D (2)jk ) s a quntet of tensor feds on E, then: DΓ(N) = (L jk, Cjk, C(2)jk) gven by (2.3) are the coeffcents of an N -near connecton D on E. The tensor feds (A j, A (2) j, B jk, D jk, D (2)jk ) are caed the dfference tensor feds of DΓ(N) to DΓ(N) and the mappng DΓ(N) DΓ(N) gven by (2.3) s caed a transformaton of N-near connecton to N-near connecton, [2]. 3 The noton of genera conforma amost sympectc N-near connecton n the bunde of acceeratons Let M be a rea n = 2n -dmensona C -manfod and et (Osc 2 M, π, M) be ts 2-oscuator bunde. The oca coordnates on the tota space E = Osc 2 M are denoted by (x, y, y (2) ). We consder on E an amost sympectc d-structure, defned by a d-tensor fed of the type (, 2), et us say a j (x, y, y (2) ), aternate: (3.1) a j (x, y, y (2) ) = a j (x, y, y (2) ), and nondegenerate:

5 Genera Conforma Amost Sympectc N-Lnear Connectons 11 (3.2) det aj (x, y, y (2) ), y, y (2). We assocate to ths d-structure Obata s operators: (3.3) Φ r sj = 1 2 ( s r j a sja r ), Φ r sj = 1 2 ( s r j + a sja r ), where (a j ) s the nverse matrx of (a j ): (3.4) a j a jk = k. Obata s operators have the same propertes as the ones assocated wth the metrca d-structure on E, [8]. Let A 2 (E) be the set of a aternate d-tensor feds of the type (, 2) on E. As s easy shown, the reaton for b j, c j A 2 (E) defned by: (3.5) b j c j { ρ(x, y, y (2) ) F(E)b j = e 2ρ c j } s an equvaent reaton on A 2 (E). Defnton 3.1 The equvaent cass: â of A 2 (E) /, to whch the amost sympectc d-structure a j beongs, s caed a conforma amost sympectc d-structure on E = Osc 2 M. Every a j â s a d-tensor fed aternate and nondegenerate, expresed by: (3.6) a j = e2ρ a j. Obata s operators are defned for a j â by puttng (a j ) = (a j ) 1. Snce equaton (3.6) s equvaent to: (3.7) (a j ) = e 2ρ a j, we have Proposton 3.1 Obata s operators depend on the conforma amost sympectc d-structure â, and do not depend on ts representatve a j â. Let N be a nonnear connecton on E wth the coeffcents (N j, N (2) j ) and et D be an N-near connecton on E wth the coeffcents n the adapted bass {,, } : DΓ(N) = (L x y y (2) jk, C jk, C (2)jk ).

6 12 Monca Purcaru and Ncoeta Adea Defnton 3.2 An N-near connecton D on E, s sad to be a genera conforma amost sympectc N-near connecton on E, f t verfes the foowng reatons: (3.8) a jk = K jk, a j k = Q jk, (α = 1, 2), where K jk, Q jk, (α = 1, 2), are tensor feds of the type (, 3), havng the propertes of antsymmetry n the frst two ndces: (3.9) K jk = K jk, Q jk = Q jk, (α = 1, 2),,, denote the h-and v α -covarant dervatves, (α = 1, 2), wth respect to DΓ(N). Partcuary, we can gve: Defnton 3.3 An N-near connecton D on E, for whch there exsts a 1-form ω n X (Osc 2 M), (ω = ω dx + ω y + ω (2) y (2) ) such that: (3.1) a jk = 2 ω k a j, a j k = 2 ω k a j, (α = 1, 2), where and denote the h-and v α -covarant dervatves (α = 1, 2) wth respect to DΓ(N), s sad to be compatbe wth the conforma amost sympectc structure â, or a conforma amost sympectc N-near connecton on E wth respect to the conforma amost sympectc structure â, correspondng to the 1-form ω, and s denoted by: DΓ(ω). For any representatve a j â we have: Theorem 3.1 For a j = e2ρ a j, a conforma amost sympectc N-near connecton wth respect to â, correspondng to the 1-form ω, DΓ(ω) satsfes: (3.11) a jk = 2 ω k a j, a j where ω = ω + dρ. k = 2 ω a j, (α = 1, 2), Snce n Theorem 3.1. ω = s equvaent to ω = d( ρ), we have:

7 Genera Conforma Amost Sympectc N-Lnear Connectons 13 Theorem 3.2 A conforma amost sympectc N-near connecton wth respect to â, correspondng to the 1-form ω, DΓ(ω), s an amost sympectc N-near connecton wth respect to some a j â, (.e. a jk =, a j k =, (α = 1, 2)), f and ony f ω s exact. 4 The set of a genera conforma amost sympectc N-near connectons n the bunde of acceeratons Let N and N be two nonnear connectons on E = Osc 2 M, wth the coeffcents ( N Let D Γ ( N) = ( j, N(2) j Ljk, Cjk, C(2)jk ) and (N j, N (2) j ) respectvey. ), be the coeffcents of an arbtrary fxed N-near connecton on E. Then any N-near connecton on E, wth the coeffcents DΓ(N) = (Ljk, C jk, C ), can be expressed n the form (2)jk (2.3), takng DΓ(N) for D Γ( N) and D Γ ( N) for DΓ(N), where (A j, A(2) j, B jk, D jk, D (2)jk ) s the dfference tensor feds of D Γ ( N) to DΓ(N). In order that DΓ(N) s a genera conforma amost sympectc N-near connecton on E, that s (3.8) hods for DΓ(N), t s necessary and suffcent that Bjk, D jk, D (2)jk satsfy: (4.1) Φ r sj Bs rk = 1 2 am [a + A mj k a mj k Φ r sj D +(A(2) k + A r s k rk = 1 2 am (a mj N r )a mj (2) K mjk ], k + A k a mj Q mjk ), Φ r sj D s (2)rk = 1 2 am (a mj k Q (2)mjk ), where and, (α = 1, 2), denote the h-and v α -covarant dervatves,

8 14 Monca Purcaru and Ncoeta Adea (α = 1, 2), wth respect to D Γ ( N). Thus, we have: Proposton 4.1 Let D Γ ( N) be a fxed N-near connecton on E. Then the set of a genera conforma amost sympectc N-near connectons, DΓ(N) s gven by (2.3), where Bjk, D, (α = 1, 2), are arbtrary tensor jk feds satsfyng (4.1). Especay, f D Γ ( N) s a genera conforma amost sympectc N-near connecton, then (4.1.) becomes: (4.2) Φ r sj Bs rk = 1 2 am [A Φ r sj D Φ r sj D s rk = 1 2 am A s (2)rk =, k a mj k a mj + (A (2) k + A r k, r )a mj N (2) ], From Theorem 5.4.3[4], however, the system (4.1) has soutons n B Djk, (α = 1, 2). Substtutng n (2.3) from the genera souton we have: jk, Theorem 4.1 Let D Γ ( N) be a fxed N-near connecton on E. The set of a genera conforma amost sympectc N-near connectons DΓ(N) s gven by:

9 Genera Conforma Amost Sympectc N-Lnear Connectons 15 (4.3) Ljk = Ljk +X k C C C k N m C j + X (2)jm + X am [a + X mj k a mj + (X(2) k + X r k N k K mjk ] + Φ r sj Xs rk, jk = Cjk + C(2)j X Q mjk ) + Φ r (2)jk = C sj Y rk s, (2)jk am (a mj k am (a mj (2) k Q (2)mjk ) + Φ r (2) k C (2)j + r )a mj (2) k + X k a mj sj Y (2)rk s, (2) where N j = N j X tensor feds, and, 1, 2), wth respect to D Γ ( N). j, X j, X jk, Y jk, (α = 1, 2) are arbtrary, denote the h-and v α -covarant dervatves, (α = If we take a genera conforma amost sympectc N-near connecton as D Γ ( N), n Theorem 4.1, then (4.3) becomes: (4.4) C where N Ljk = Ljk +X k C j + X k N m C am [X k Q mj + (X (2) k + X r k (2)jm + X (2) k C (2)j + N r )Q (2)mjk] + Φ r jk = Cjk + C(2)j X k am Q (2)mj X k + Φr sj Y rk s, C (2)jk = C (2)jk + + Φr j = N j X tensor feds and, wth respect to D Γ ( N). sj Y (2)rk s, j, X j, X jk, Y jk sj Xs rk,, (α = 1, 2) are arbtrary, denote the h-and v α -covarant dervatves, (α = 1, 2),

10 16 Monca Purcaru and Ncoeta Adea Observatons 4.1. () If we consder X j = X jk = Y jk =, (α = 1, 2), then from (4.3) we obtan the set of a genera conforma amost sympectc N-near connectons, correspondng to the same nonnear connecton N, [13]. () If we take K jk = 2a j ω k, Q jk = 2a j ω k, (α = 1, 2), such that ω = ω dx + ω y + ω (2) y (2) s a 1-form n X (Osc 2 M), and f we preserve the nonnear connecton N, (.e.n = N), then from (4.3) we obtan the set of a conforma amost sympectc N-near connectons, correspondng to the same nonnear connecton N, [12]. () If we consder K jk =, Q jk =, (α = 1, 2), and f we preserve the nonnear connecton N, (.e.n = N), then from (4.3) we obtan the set of a amost sympectc N-near connectons, correspondng to the same nonnear connecton N, [11]. (v) Fnay, f we preserve the nonnear connecton N, (.e.n = N) from (4.4), we obtan the transformatons of genera conforma amost sympectc N-near connectons, correspondng to the same nonnear connecton N, [13]. References [1] Atanasu Gh., Ghnea I.Connexons Fnserennes Généraes Presque Sympectques, An. Şt.Unv. A.I.Cuza Iaş, Secţ I a Mat. 25 (Sup.), 1979, [2] Matsumoto M.,The theory of Fnser connectons,pub.study Group Geom.5, Depart.Math., Okayama Unv., 197. [3] Mron R.,Introducton to the Theory of Fnser Spaces,The Proc.of Nat.Sem. on Fnser Spaces, Braşov, 198, [4] Mron R.,The Geometry of Hgher-order Lagrange spaces. Appcatons n Mechancs and Physcs,Kuwer Academc Pubshers, FTPH82, [5] Mron R. and Atanasu Gh.,Lagrange Geometry of Second Order, Math. Comput. Modeng,Pergamon, 2, 4/5(1994), [6] Mron R. and Atanasu Gh.,Proongaton of Remannan, Fnseran and Lagrangan structures, Rev.Roumane Math.Pures App., 41,3/4(1996),

11 Genera Conforma Amost Sympectc N-Lnear Connectons 17 [7] Mron R. and Atanasu Gh.,Hgher-order Lagrange spaces, Rev. Roumane Math.Pures App., 41,3/4(1996), [8] Mron R. and Hashguch M.,Metrca Fnser Connectons, Fac.Sc., Kagoshma Unv.(Math., Phys.& Chem.), 12(1979), [9] Mron R. and Hashguch M., Conforma Fnser Connectons, Rev.Roumane Math.pures App., 26, 6(1981), [1] Purcaru M., Şofroncu I., Păun M., The Amost Sympectc Structures n the Bunde Space of the Acceeratons,Proc. of the 23 rd Conf. on Geometry and Topoogy, 1993, [11] Purcaru M.,Amost Sympectc N-Lnear Connecton n the Bunde of Acceeratons,Nov Sad J.Math., vo.29, no.3(1999), [12] Purcaru M., Stoca E., Conforma Amost Sympectc N-Lnear Connectons n the Bunde of Acceeratons,Proc. of the Fourth Internatona Workshop on Dfferenta Geometry and Its Appcatons, Braşov, sept. 1999, [13] Purcaru M., Stoca E., Târnoveanu M., Nondegenerate Antsymmetrca Structures n the Bunde of Acceeratons,Bu.Maaysan Math.Sc.Soc. (Second Seres), 25(22), Transvana Unversty of Braşov, Department of Geometry, Iuu Manu 5, 22 Braşov, Romana E-ma: mpurcaru@untbv.ro