AALELE ŞTIIŢIFICE ALE UIVERSITĂŢII AL.I. CUZA DI IAŞI (S.. MATEMATICĂ, Tomul LIII, 2007, Suplment ATURAL 2-π STRUCTURES I LAGRAGE SPACES Y VICTOR LĂUŢA AD VALER IMIEŢ Dedcated to Academcan Radu Mron at hs 80th annversary Abstract. We nvestgate the concepts of almost 2-π structure, almost 2-π Remannan structure, almost symplectcal 2-π structure on T M and n-almost contact 2-π structure, n-almost contact Remannan structure on Osc 2 M. These structures clearly exst n the natural case when they are gven by (1.1, (2.6, (3.10, (4.1. The theorems 1.1, 2.2, 3.1 and 4.1 pont out some mportant propertes. We nvestgate also the condtons of normalty for these structures. Mathematcs Subject Classfcaton 2000: 53C60, 5340. Key words: 2-π structure, Lagrange space, almost contact structures. Introducton. In the geometry of Lagrange spaces L n = (M, L one ntroduces n a natural way an almost complex structures F nduced by a canoncal nonlnear connecton. Then whenever one consders the Lagrange spaces of order 2, L (2n = (M, L the prevous structure become more general, that s, an almost contact stucture F, also generated by a canoncal nonlnear connecton whch together wth the Sasakan lft G of fundamental metrc tensor of space defne a natural almost contact metrc structure denoted by H 3n = (Osc 2 M, G, F whch represents, the geometrcal model of background space. In our further consderatons, we shall prove that above the knds of structures can be extend to 2-π structures studed by us n the works [1], [2], [3], [4]. Moreover, n the case of spaces L n = (M, L we shall prove that there exsts a natural 2-π structure nduced by the canoncal nonlnear connecton
74 VICTOR LĂUŢA AD VALER IMIEŢ 2 of the space, and the structure together wth Sasakan s lft G of the fundamental tensor generates a metrcal 2-π structure on the total space of the tangent bundle T M. Also, we shall show that the metrc canoncal lnear connecton of the space L n s compatble wth the correspondng structures F and G. Fnally we shall prove that the almost symplectcal structure assocated s ntegrable. In general for Lagrange spaces of order 2, L (2n we shall show that there exsts an almost contact natural 2-π structure F generated by the nonlnear canoncal connecton. Afterwards, we prove that any -lnear connecton D s compatble wth the structure F, that s, DF = 0 and by consderng the lft of Sasak s type G for the fundamental tensor of the space, we show that the par (F, G s an almost contact metrcal 2-π structure. The normalty of ths structure s characterzed. 1. Almost 2-π structure on T M. Let M be a real n-dmensonal dfferentable manfold of the class C and (T M, Π, M ts tangent bundle. We denote by (x, y (, j, h = 1,..., n the canoncal coordnates of every pont u = (x, y T M. The local coordnate transformatons (x, y ( x, ỹ on T M are usually gven by (1.1 x = x (x 1,..., x n, det x x j 0 ỹ = x x j yj. The Louvlle vector feld Γ s defned by Γ = y y and the vertcal dstrbuton V on T M s locally generated by the system of vector felds ( y. It s known that a nonlnear connecton s a regular dstrbuton on T M such that we have the followng drect sum T u (T M = u V u for any pont u T M. There exsts a local adapted bass of dstrbuton and V gven by { x, y } ( = 1,..., n where (1.2 x = x j y j. The system of the functons j are the coeffcents of nonlnear connecton.
3 ATURAL 2-π STRUCTURES I LAGRAGE SPACES 75 Let {dx, y } ( = 1,..., n be the dual bass of adapted bass. We have (1.2 y = dy + j dxj. Defnton 1.1. [2] An almost 2-π structure on T M s defned as an F(T M-lnear mappng F : χ(t M χ(t M wth the property (1.3 F F = λ 2 I, where λ 0 s a complex number, I beng a Kronecker tensor. One says that F s ntegrable f the jeuhus tensor F = 0, F beng gven by (1.4 F (X, Y = F 2 [X, Y ] + [FX, FY ] F[FX, Y ] F[X, FY ]. We have { } Theorem 1.1 If s a nonlnear connecton on T M and, x y s the adapted bass of dstrbuton then the F(T M-lnear mappng F : χ(t M χ(t M defned by (1.5 F ( x = λ y, F s an almost 2-π structure on T M. ( y = λ ( = 1, 2,..., n x Proof. From (1.5 one sees that the structure F s well defned that ths s t does not depend on the transformaton of coordnates (1.1 on T M. Then (1.5 mples ( (1.6 F 2 x = λ 2 ( x, F2 y = λ 2 y. Theorem 1.2. The almost 2-π structure F gven by (1.5 s ntegrable, f and only f, the torson Tjh and the curvature tensor R jh of the nonlnear connecton are equal wth zero, where (1.7 T jh = j y h h y j, R jh = j x h h x j.
76 VICTOR LĂUŢA AD VALER IMIEŢ 4 Proof. Snce we have [ ] [ ], x j x = R h jh,, = j through y x j y h y h y the agency of jeuhns tensor s relaton (1.4 we obtan ( ( ( F x j, x h = λ 2 Rjh y T jh x, F x j, y h (1.8 ( = λ (T 2 jh y R jh x, F y j, y h = λ 2 Rjh y whch prove that Tjh = 0, R jh = 0 mples F = 0 and recprocally. In all our further consderaton the ntegrable almost 2-π ntegrable structure wll be called 2-π structure. 2. Metrcal almost 2-π structure on T M. Let L n = (M, L be a Lagrange space. The Lagrangan L s the fundamental functon of the space and (2.1 g j (x, y = 1 2 2 L y y j s the fundamental tensor of the space L n = (M, L [7]. From the defnton of Lagrange space we have that (2.2 rank g j (x, y = n. We can consder the contravarant g j (x, y of the fundamental tensor. We have n [6] the followng result: Theorem 2.1. For any Lagrange space L n there exsts a nonlnear connecton dependng only of the fundamental functon L havng the coeffcents gven by (2.3 j = G y j where (2.3 G = 1 4 gj ( 2 L y x h yh L x j Wth these coeffcents the connecton s called the canoncal connecton (f the space L n s a Fnsler space F n, then t concdes wth famous Cartan connecton..
5 ATURAL 2-π STRUCTURES I LAGRAGE SPACES 77 The adapted bass of dstrbuton and V s defned by ( x, y wth from (1.2, j x gven by (2.3 and (2.3 and the dual bass (dx, y defned n (1.2 wth the same coeffcents j of canoncal nonlnear connecton. Then, the Sasakan lft G of the fundamental tensor has the followng expresson (2.4 G = g j dx dx j + g j y y j. We have the followng property: Theorem 2.2. The par (F, G s an almost Remannan 2-π structure. Proof. It s necessary to verfy the relaton G(FX, FY = λ 2 G(X, Y, X, Y χ(t M. We have ( G F x, F x j ( G F x, F y j ( G F y, F x j ( = λ 2 G ( = λ 2 G ( = λ 2 G x, From the book [6] we have ( y, y j = λ 2 g j = λ 2 G x, x j ( y, x j = 0 = λ 2 G x, y j ( x j = λ 2 g j = λ 2 G y, y j Theorem 2.3. Any metrcal -connecton D wth h- and v-metrc torson zero has the coeffcents gven by the generalzed symbols of Chrstoffel L jh = 1 ( gmh 2 gm x j + g jm x h g jh x m (2.5 Cjh = 1 ( gmh 2 gm y j + g jm y h g jh y m. It follows that we have h-and v-covarant dervatves of the fundamental tensor equal wth zero g j h = 0, g j h = 0 whch mples that D X G = 0 for all X χ(t M. One check that D X F = 0.
78 VICTOR LĂUŢA AD VALER IMIEŢ 6 ow, we consder almost smplectcal 2-π structure Θ assocated to the structure (F, G and defned by (2.6 Θ(X, Y = G(FX, Y. We have ( ( ( Θ x, x j = 0, Θ x, y j = λg j, Θ y, y j = 0. Therefore (2.7 Θ = λg j y dx j. Theorem 2.4. For any Lagrange space, almost symplectcal 2-π structure Θ s ntegrable. Proof. The almost symplectcal 2-π structure s ntegrable, f and only f ts exteror dfferental s equal wth zero. ecause, any Lagrange space L n one has the property DΘ 1 = 0 where Θ 1 = g j y dx j (see [7] t follows that Θ = λθ 1, has the same property. Remark 2.1. It s clear that the above theory remans vald whenever the space L n s replaced by the Fnsler space F n and t can be extended to the study of the general almost 2-π structure on tangent bundle [1],[2],[3],[4]. 3. n-almost contact 2-π structure on OSC 2 M. The extenson of the above theory to the case of the 2-osculator bundle s not mmedate because the almost 2-π structures are not specfcally for the dfferentable manfolds Osc 2 M and, n general, ther exstence s not ensured. We shall demonstrate n ths secton that on Osc 2 M endowed wth a nonlnear connecton there exsts a natural n-almost contact 2-π structure compatble also wth a Sasakan metrc n the case of Lagrange spaces of order 2. Frst of all, we need some prelmnary consderatons. Let us consder agan M be a real n-dmensonal dfferentable manfold and (Osc 2 M, Π, M ts 2-osculator bundle. It s know from [6] that the ponts of Osc 2 M are osculator spaces of order 2 of manfold M.
7 ATURAL 2-π STRUCTURES I LAGRAGE SPACES 79 We denote by (x, y (1, y (2 the canoncal coordnates for any pont u = (x, y (1, y (2 Osc 2 M. Then, the canoncal coordnate transformatons (x, y (1, y (2 ( x, ỹ (1, ỹ (2 are gven by (3.1 x = x (x 1,..., x n, rank x x j = n, ỹ(1 = x 2ỹ (2 = ỹ(1 x j y(1j + 2 ỹ(1 y(2j y (1j x j y(1j and there exsts a tangent structure J : χ(e χ(e on the manfold E = Osc 2 M defned by (3.2 J ( x = ( y (1, J y (1 = ( y (2, J y (2 = 0. On Osc 2 M there exsts the vertcal dstrbutons V 1, V 2 wth the propertes V 1 V 2, dm V 1 = 2n, dm V 2 = n [6]. All these dstrbutons are ntegrable and there exsts a number of two Louvlle vector felds lnearly ndependent Γ, 1 Γ. 2 We have the followng propertes: rank J = 2n, ImJ = V 1, kerj = V 2, ImJ V1 = V 2, ImJ V2 = 0 and J 2 Γ= 1 Γ, 0 Γ= 0, J s an ntegrable structure. A 2-spray on Osc 2 M [6] s a vertcal feld S on Osc 2 M wth the property JS = 2 Γ. Then S has form (3.3 S = y (1 x + 2y(2 y (1 3G (x, y (1, y (2 y (2. In [6] one proves that the fundamental geometrcal objects lke as the nonlnear connectons, -lnear connectons and so on are generated by a 2-spray and any nonlnear connecton on Osc 2 M s defned now as a regular dstrbuton 0 on Osc 2 M supplementary of the vertcal dstrbuton. Wrtng 1 = J( 0, V 2 = J( 1 for any nonlnear connecton 0 one obtans the decomposton n drect sum of tangent bundle at E = Osc 2 M gven by (3.4 T u E = 0 (u 1 (u V 2 (u.
80 VICTOR LĂUŢA AD VALER IMIEŢ 8 All the geometrcal objects on Osc 2 M are consdered wth respect to ths drect sum and the correspondng locally adapted bass to the decomposton (3.4 s ( (3.5 x, y (1, y (2, where (3.6 x = x (1 m y (1m (2 m y (2m, y (1 = y (1 (1 m y (2m, y (2 = y (2 n whch the system of functons j(1, j(2 gve the coeffcents of the nonlnear connecton 0. The dual bass of (3.5 s defned by (3.7 {x, y (1, y (2 } where (3.8 x = dx, y (1 = dy (1 + M j(1 dxj, y (2 = dy (2 + M j(1 dy(1 + M j(2 dxj. The coeffcents Mj(1, M j(2 are named the dual coeffcents of nonlnear connecton 0 and there exsts a relatonshp of dualty between the drect coeffcents and dual coeffcents for any nonlnear connecton. In [6] one ndcates a method of calculus for dual coeffcents when a 2-spray s gven. We can defne now the natural n-almost contact 2-π structure. Defnton 3.1. A k-almost contact 2-π structure on E = Osc 2 M s a system (3.9 (F, ξ 1,..., ξ k, η 1,..., η k, where F s a tensor feld of type (1.1 on Osc 2 M = Osc 2 M \ {0}, ξ 1,..., ξ k are the lnear ndependent vector felds and η 1,..., η k are 1-form felds, such that we have k F 2 (X = λ 2 X + λ 2 η a (Xξ a, X χ(e (3.10 a=1 F(ξ a = 0, η a (ξ b = b a, (a, b = 1,..., k
9 ATURAL 2-π STRUCTURES I LAGRAGE SPACES 81 where λ s a non zero complex number. Then t follows that ξ 1,..., ξ k belong to the kernel of the mappng F and we have (3.11 F 3 + λ 2 F = 0. The jeuhns tensor s gven by the relaton (1.4 and the normalty condton s n (3.12 F (X, Y + λ 2 dη a (X, Y = 0, X, Y χ(e. a=1 We prove that n a Lagrange space L (2n = (M, L there exsts n-almost contact 2-π structure. We have Theorem 3.1. Let be a nonlnear connecton on Ẽ = Osc ( 2 M havng the adapted bass. On Ẽ there exsts an n-almost contact,, x y (1 y (2 2-π structure (3.9 when F s the F-lnear mappng defned by ( F x = λ y (2, ( (3.13 F = 0, and F y (1 ( y (2 = λ x (3.14 ξ a = y (1a, ηa = y (1a (a = 1, 2,..., n. Proof. From (3.13 one remark that F s globally defned on Ẽ. We verfy the relaton (3.10 n the base of relatons (3.13 and (3.14. Clearly, the second and the thrd lnes of (3.10 s vald because ξ a from (3.14 belong to Ker F and η a are ther duals. Let us verfy the frst relaton from (3.10. Indeed, we have ( ( F 2 x = F λ = λ 2 x y (1
82 VICTOR LĂUŢA AD VALER IMIEŢ 10 or equvalently ( F 2 x = λ 2 n [ ( ] x + y a x ξ a a=1 because the last sum s zero. In the same way one verfes (3.10 for χ = and χ = y (1 y (2 The n-almost contact 2-π structure wll be called natural. The above theorem leads mmedately to. q.e.d. Corollary 3.1 The natural n-almost contact 2-π structure has the followng propertes: 1 0. F s a tensor feld of type (1.1 globally defned on Ẽ. 2 0. KerF = 1, ImF = 0 V 2. 3 0. rank F = 2n 4 0. F 3 + λ 2 F = 0 The problem whch must solved now s to specfy condtons of normalty for ths structure. Frst of all we have Theorem 3.2. The natural n-almost contact 2-π structure (3.9 s normal f and only f we have n (3.15 F (X, Y + λ 2 d( (1 y a (X, Y = 0. a=1 Proof. One apples (3.12 n whch η a = y (1a (a = 1,..., n and consder X and Y equals wth the vectors of the adapted bass. We obtan Theorem 3.3. The almost n-contact 2-π structure (F, ξ a, η a s normal f and only f, the followng equatons holds (3.16 R (12 (21 jk = jk (11 (21 jk = kj (12 jk = 0; (01 R jk = 0; (22 jk (22 kj R (02 jk = 0
11 ATURAL 2-π STRUCTURES I LAGRAGE SPACES 83 where (3.17 R (12 jk = (1 j y (1k (1 k y (1j, R (01 jk = (1 x k R (02 jk = (1 m (21 jk = (1 j y (2k ; (22 R (01 jk + (2 x k (11 jk = (1 j y (1k ; (12 j (1 jk = jk = (2 x j m (1 m k m j (1 k x j ; (11 jk + (2 j y (1k (2 k y (1j ; m (21 jk + (2 j y (2k. Our above theoretcal consderatons prove the valdty of the followng theorem: Theorem 3.4. If the basc manfold M s paracompact, then there exsts on the total space of ts 2-osculator bundle at least one n-almost contact 2-π structure. Ths theorem allows us to defne on Osc 2 M the more general n-almost contact 2-π structure and to study them usng the methods presented n ths secton. 4. n-almost contact remannan 2-π structure. Let us consder a k-almost contact 2-π structure on the Lagrange space L (2n gven by (F, ξ 1,..., ξ k, η 1,..., η k and G a Remannan structure on E = Osc 2 M. Defnton 4.1. One calls k-almost contact Remannan 2-π structure any k-almost contact 2-π structure together wth a Remannan structure G on Ẽ for whch n G(F X, F Y = λ 2 G(X, Y G(X, λ 2 η a (Xξ a (4.1 a=1 G(X, ξ a = η a (X, (a = 1,..., n such that the base (η a s orthonormal.
84 VICTOR LĂUŢA AD VALER IMIEŢ 12 ow we consder a Lagrange space n whch L(x, y (1, y (2 s the fundamental functon and (4.2 g j = 1 2 2 L y(2j y (1 s ts fundamental tensor. We have rank g j and f we consder ts contravarant g j then the Sasakan lft G of the fundamental tensor g j s gven by (4.3 G = g j dx dx j + g j y (1 y (1j + g j y (2 y (2j, where dx, y (1, y (2 s the dual bass of the adapted bass for the nonlnear canoncal connecton of the space L (2n. Followng the same lne as n the usual cases of the Remannan metrc structure n the Lagrange spaces L (1n presented n secton 1 and 2 we obtan the next result: Theorem 4.1. The natural n-almost contact 2-π structure (F, ξ 1,..., ξ n, η 1,..., η n together wth the Sasakan lft G gven by (4.3 s a n-almost contact Remannan 2-π structure on Osc 2 M. Let us consder the metrc -connecton wth respect to G whch has the coeffcents gven by generalzed Chrstoffel symbols: (4.4 L m j = 1 ( gs 2 gms x j + g sj x g j x s Cj(α m = 1 ( gs 2 gms y (αj + g sj y α Therefore we have g j y (αs α = 1, 2. Theorem 4.2. If D s canoncal -connecton wth the coeffcents (4.4 then natural n-almost contact Remannan 2-π structure (F,G has the property D X F = 0 and D X G = 0 X χ(osc 2 M. Remark 4.1. The prevous theory can be partcularzed n the case of normal n-almost contact Remannan 2-π structure. The prevous theory proves that on the dfferentable paracompact manfold M and Osc 2 M there exsts n-almost contact Remannan 2-π structures and that ther general theory can be studed through the agency the specfed methods.
13 ATURAL 2-π STRUCTURES I LAGRAGE SPACES 85 Thus n general, the bass problems of the n-almost contact Remannan 2-π structures n Lagrange spaces of order 2 are elucdated. All ths theory can be appled n the partcular case of the Fnsler spaces of order 2 by mposng on the Lagrangan certan condtons of homogeneousness. REFERECES 1. lanuţa, V.; Yawata, M. Infntezmal transformatons of the 2-π structures on tangent bundle,.s. Tensor Japan 55 (1994, 43-52. 2. lanuta, V.; Yawata, M. Infntezmal transformatons of metrcal 2-π structures, Tehn. Inst. of Chba, Japan, 41(1994, 17-24. 3. lanuţa, V. atural 2-π structures n Lagrange spaces of hgher order, Journal of the Egyptan Mathematcal Socety, Vol 9(2, 2001, 151-163. 4. lanuţa, V.; Grtu, M. atural n-almost contact 2-π structures n Lagrange spaces of order 2, Stud s Cercetar Stntfce, Sera Matematca, Unverstatea dn acau, 16(2006, 3-10. 5. lanuţa, V.; mneţ, V. Metrcal n-almost contact 2-π structures n Lagrange spaces of order 2, Stud s Cercetar Stntfce, Sera Matematca. Proceedng of nternatonal Conference of Mathematcs ICMI45 acau, Sept.2006, 75-84. 6. Mron, R. The Geometry of Hgher Order Lagrange Spaces, Applcatons to Mechancs and Physcs, FTPH Vol. 82(1997, Kluwer Acad. Publ. Holland. 7. Mron, R.; Anastase, M. The Geometry of lagrange Spaces, Theory and Applcatons, FTPH vol.59(1994, Kluwer Acad. Publ. Holland. Receved: 15.X.2007 Unversty of acău, Depart. of Mathematcs and Informatcs, ROMÂIA vblanuta@ub.ro valern@ub.ro