SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS IN THE 2-TANGENT BUNDLE

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1 STUDIA UNIV. BABEŞ BOLYAI MATHEMATICA Volume LIII Number March 008 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS IN THE -TANGENT BUNDLE GHEORGHE ATANASIU AND MONICA PURCARU Abstract. In the present paper we study the transformatons for the coeffcents of an N-lnear connecton (defnton. on the tangent bundle of order two T M by a transformaton of a nonlnear connecton n T M. We prove that the set T of these transformatons together wth composton of mappngs sn t a group. But we gve some groups of T whch keep nvarant a part of components of the local coeffcents of an N-lnear connecton. We also determne the transformaton laws of the torson and curvature tensor felds wth respect to the transformatons of the group T N T.. The N-and JN-lnear connectons on tangent bundle of order two Let M be a real C -manfold wth n dmensons and (T M π M ts - tangent bundle []. The local coordnates on 3n-dmensonal manfold T M are denoted by (x y ( y ( = (x y ( y ( = u ( =...n. ( Let x y ( be the natural bass of the tangent space T T M at y ( the pont u T M and let us consder the natural -tangent structure on T M J : χ(t M χ(t M gven by ( J x = ( y J = ( ( y ( y J = 0. (. ( y ( (N j N j We denote wth N a nonlnear connecton on T M wth the local coeffcents ( j =... n [7] [8]. Receved by the edtors: Mathematcs Subject Classfcaton. 53C05 53C5 53C 53C60 53B05 53B35 53B40. Key words and phrases. -tangent bundle N-lnear connecton JN-lnear connecton transformatons group subgroup torson curvature. 3

2 GHEORGHE ATANASIU AND MONICA PURCARU Hence the tangent space of T M n the pont u T M s gven by the drect sum of the lnear vector spaces: T n T M = N 0 (u N (u V (u u T M. (. An adapted bass to the drect decomposton (. s gven by { } x y (.3 ( y ( where: x = x N j = y ( y ( =. y ( y ( y (j y (j N j Let us consder the dual bass of (.3: N j y (j (.4 {dx y ( y ( } (.5 where x = dx y ( = dy ( + N jdx j y ( = dy ( + N jdy (j + (N j + N m N m j dx j. (.6 Defnton.. ([]-[3] A lnear connecton D on T M D : χ(t M χ(t M χ(t M s called an N-lnear connecton on T M f t preserves by parallelsm the horzontal and vertcal dstrbutons N 0 N and V on T M. An N-lnear connecton D on T M s characterzed by ts coeffcents n the adapted bass (.3 n the form: 4 D x k x j D D y (k x j y (k x j = L (0 (0 x D x k x D x D y (j y (k y (j y (k y (j = L (0 ( ( y ( D x k y ( D y ( D y (j = L (0 y (k y (j y (k y (j y ( ( y ( y (. (.7

3 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS The system of nne functons DΓ(N = ( L L (0 L (0 (0 ( ( (0 are called the coeffcents of the N-lnear connecton D. ( C (.8 Generally an N-lnear connecton DΓ(N on T M s not compatble wth the natural -tangent structure J gven by (.. Defnton.. An N-lnear connecton D on T M s called JN-lnear connecton f t s absolute parallel wth respect to D: D X J = 0 X χ(t M. (.9 Theorem.. (Gh. Atanasu [] A JN-lnear connecton on T M s characterzed by the coeffcents JDΓ(N gven by (.8 where L = L (0 = L (0 (= L C (0 ( ( ( ( (.0 C (0 ( (. ( coeffcents: It results that a JDΓ(N- lnear connecton on T M has three essentally JDΓ(N = (L C ( C. (. ( Obvous the geometrcal theory on -tangent bundle (T M π M wth the N- lnear connecton []-[3] [5] generalze on that wth the JN-lnear connecton (cf.wth R. Mron and Gh. Atanasu [5]-[8]; see also M. Purcaru [] [3]. In the followng we use the N-lnear connectons only. 5

4 GHEORGHE ATANASIU AND MONICA PURCARU. The set of transformatons of N-lnear connectons Let DΓ(N be an N-lnear connecton on T M wth the coeffcents gven by (.8. If N s another nonlnear connecton on T M wth the coeffcents ( N j A j N j ( j =... n then there exsts the unquely determned tensor felds τ (T M such that: Conversely f N j nonlnear connecton. x = C x = t follows: 6 D x k N j and A j = N j A j ( =. (. are fxed then N j ( = gven by (. s a Let us suppose that the mappng N N s gven by (. and we denote: x N j y N j (j y (j y ( = y N j ( y (j y ( = y (. It follows frst of all that the transformatons (. preserve the coeffcents ( = 0. Takng n account the fact that: x + Aj = D y (j x k = L (0 D = D x k y (k = D y (k y + (j Aj y + (j Al k y + ( Al k = ( L (0 y (j = D y (j y (j = L (0 Cjl ( y (k y + (j Al k y ( = y = D ( ( D ( y (l +N m l y (m y + ( Al k C jl ( y (j = D y (l ( N m l N m l y + ( Aj x k +Al k y + (j Al k C jm C jm y (j y (l +Al k D y (l y + ( Al k y = D ( ( y (j = C ( C jl y (k +Al k y + ( Al k y ( = y (l y (j = C jl y. ( y (l C jl y ( y = (j y ( = y = (j y ( =

5 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS = ( C ( C jl y. ( Therefore the change we are lookng for s: L = L C jl ( ( ( C jl ( = 0. N m l C jm C jl (. So we have proved: Proposton.. The transformaton (. of nonlnear connectons mply the transformatons (. for the coeffcents DΓ(N of the N-lnear connecton D. Theorem.. Let N and N be two nonlnear connectons wth the coeffcents (N j N j ( N j N j-respectvely. If DΓ(N = ( L L (0 L (0 (0 ( ( (0 ( C and D Γ( N = ( L L (0 L (0 (0 ( ( (0 ( are two N- respectvely N-lnear connectons on the dfferentable manfold T M then there exsts only one system of tensor felds (A j A j (0 (0 (0 ( ( (0 ( such that: N j L ( = N j A j = L ( C jl ( C jl ( N m l C jm ( = 0 ; =. C jl B (.3 7

6 GHEORGHE ATANASIU AND MONICA PURCARU Proof. The frst equalty (.3 determnes unquely the tensor felds A j ( =. Snce C C C C C and are tensor felds the second equaton (0 ( ( (0 ( C (.3 determnes unquely the tensor felds B B (0 and the fourth equaton (.3 determne the tensor felds (0 respectvely. ( Conversely we have Theorem.. If and B (0 (0. Smlarly the thrd ( ( and DΓ(N = ( L L (0 L (0 (0 ( ( are local coeffcents of an N-lnear connecton D and (0 ( C (A j A j (0 s a system of tensor felds then: (0 (0 ( ( (0 ( D Γ( N = ( L L (0 L (0 (0 ( ( (0 ( gven by (.3 are local coeffcents of an N-lnear connectons D. The system of tensor felds (A j A j (0 (0 (0 ( ( (0 ( are called the dfference tensor felds of DΓ(N to D Γ( N and the mappng DΓ(N D Γ( N gven by (.3 s called the transformaton N-lnear connecton and t s noted by: of N-lnear connecton to t(a j A j (0 (0 (0 ( ( (0 ( Theorem.3. The set T of the transformatons of N-lnear connectons to N- lnear connectons together wth the composton of mappngs sn t a group. Proof. Let 8 t(a j A j (0 (0 (0 ( ( (0 (. : DΓ(N D Γ( N

7 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS and t(ā j Ā j B (0 (0 (0 ( ( (0 ( be two transformatons from T gven by (.3. From (.3 we have: : D Γ( N D Γ( N We obtan: L ( = L + C jl ( D jl ( + D jl ( N j + C jl ( (A l k Ā l k Ā l k ( = N j (A l k (A j + Āl k + ( C jm + D jm + C jl ( B ( (A l k + Ā j ( =. + Āl k + C jm + D jm N m la l k + C jm + B + Āl k ( D jl + N m l(a l k A m l A m l ( Ā l k ( = 0. Ā l k Ā l k+ + ( + Āl k+ + ( (.4 So L ( = 0 hasn t the form (.3. Result that the mappng of two transformatons from T sn t a transformaton from T so T together wth the composton of mappngs sn t a group. Remark.. If we consder A j = 0 ( = n (.3 we obtan the set T N of transformatons of N-lnear connectons havng the same nonlnear connecton N: T N = {t(0 0 We have: (0 (0 Theorem.4. The set T N (0 ( ( (0 ( T }. of the transformatons of N-lnear connectons to N-lnear connectons together wth the composton of mappngs s a group. Ths group T N acts effectvely and transtvelly on the set of N-lnear connectons. 9

8 GHEORGHE ATANASIU AND MONICA PURCARU Proof. Let t(0 0 (0 (0 (0 ( ( (0 ( : DΓ(N D Γ(N be a transformaton from T N gven by (.5: N j = N j L ( = L ( B ( ( = 0 ; =. (.5 T N gven by: The composton of two transformatons from T N s a transformaton from t(0 0 B t(0 0 (0 (0 (0 ( ( (0 ( (0 (0 (0 ( ( (0 ( = t(0 0 B + B (0 + (0 B (0 + (0 (0 + (0 ( + ( ( + ( (0 + (0 ( + ( + The nverse of a transformaton from T N s the transformaton:. t(0 0 (0 (0 (0 ( DΓ(N D Γ(N. ( (0 ( The transformaton (.5 preserves all the N-lnear connectons D f : B = = 0 ( = 0. Therefore T N acts effectvely on the set of N-lnear connectons. From the Theorem.. results that T N acts transtvely on ths set. 0

9 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS Let us consder: T NL = {t( T N C( = {t(0 0 T N C( = {t(0 0 (0 ( (0 (0 (0 T N C( C ( = {t(0 0 (0 ( ( (0 (0 (0 (0 ( ( ( ( T N } T N } T N } T N }. Proposton.. T NL T N C( T N C( and T N C( C ( are abelan subgroups of T N. Proposton.3. The group T N preserves the nonlnear connecton N; T NL preserves the nonlnear connecton N and the components L ( = 0 of the local coeffcents DΓ(N; T N C( preserves the nonlnear connecton N and the com- preserves the non- ponents C ( = 0 of the local coeffcents DΓ(N; T N C( ( lnear connecton N and the components C ( = 0 of the local coeffcents DΓ(N and T N C( C ( preserves the nonlnear connecton N and the components C ( and C ( = 0 of the local coeffcents DΓ(N. 3. The transformatons of the d-tensors of torson and curvature n T N In the followng we shall study the Abelan group T N. Its elements are the transformatons t : DΓ(N D Γ(N gven by N j L ( = N j = L ( B ( ( = 0 ; =. (3. Frstly we shall study the transformatons of the d-tensors of torson of DΓ(N (see (7. and (7.5 []. We obtan:

10 GHEORGHE ATANASIU AND MONICA PURCARU Proposton 3.. The transformatons of the Abelan group T N gven by (3. lead to the transformatons of the d-tensors of torson n the followng way: R (0 = R (0 (3. P ( ᾱ T (0 ᾱ S ( T (0 = S ( = Q ( ᾱ Q ᾱ P ( P (0 = P ( +( B kj +( D kj ( = Q ( = S ( = Q = S ( P ( = P (0 P ( = P ( B ( ( + D kj ( + B kj (3.3 (3.4 (3.5 (3.6 (3.7 (0 (3.8 (3.9 (3.0 ( = 0 ; =. (3. Now we shall study the transformatons of the d-tensors of curvature of DΓ(N (see (7.[]. We get: Proposton 3.. The transformatons of the Abelan group T N gven by (3. lead to the transformatons of the d-tensors of curvature n the followng way: R (0 = R (0 D hs ( R s (0 D hs R s (0 B hs T s + (3. (0 P ( = P ( +A { B hj k D hs ( P s ( + B s hj D hs P s ( B sk } B hs C s ( + (3.3

11 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS P ( Q ( + L s kj D hs ( D s hk ( = P ( + L s kj B hj B sj D hs ( D hs D s hk = Q ( + D hk S ( + D s hj ( ( j ( k + C hs ( P s ( B hj B sj C s + D s hj ( = S ( D sk ( +D hk j ( B s kj D hs ( k + C hs D hs ( D sk D hs ( } D hs + B s hj D hs ( P s +D hk j B s kj + C s kj ( R s ( where A j denotes the alternate summaton. We shall consder the tensor felds: D s hk S s ( B s kj B hs + B s hj D hs D hs D sj ( B s kj D hj ( D sk ( C s D sk D hs +A { D hj ( ( k P s ( ( k ( = 0 ; = + (3.4 + (3.5 + (3.6 K h (0 = R (0 C hs ( R s (0 C hs R s (0 (3.7 P h ( = A { P ( C hs ( N s j y (k C hs (N s m N m j + (3.8 y (k P h ( = A { P ( C hs ( N s j + y (k N s k } y (j N s j y (k C hs (N s m N m j N s j y + + } (3.9 (k y (k S h ( Q h ( = S ( = A { Q ( C hs R s ( + C hs N s j } (3.0 y (k ( = 0 ; =. (3. 3

12 GHEORGHE ATANASIU AND MONICA PURCARU Proposton 3.3. By a transformaton of the Abelan group T N gven by (3. the tensor felds: K h P h P h Q h (0 ( ( ( 0 ; = are transformed accordng to the followng laws: K (0 = K (0 B hs T s +A { B hj k (0 + B s kj and S ( B sk ( = } (3. P ( D hj k ( = P ( + B s hj D hs ( D sk ( T s B hs (0 + D s hj ( B sk S s ( + D hs ( +A { B hj B s C hs ( ( k (3.3 B s } S ( Q +A { D hj ( ( = S ( ( = 0 ; =. ( k D hs ( = Q h + S s ( ( +D hk S s ( ( j + D hs ( + D s hj ( +A { D hj ( S s ( D sk ( k D hs D s hk + D s hj ( D sj ( D sk ( } (3.4 } (3.5 The transformatons for the coeffcents of an N-lnear connecton on the tangent bundle of order two T M by a transformaton of a nonlnear connecton n T M together wth the transformaton laws of the torson and curvature tensor felds wth respect to the transformatons of the group T N gven n the present paper are necessary for the study of a mportant subgroup of the group T N : the group of transformatons of the metrc sem-symmetrc N-lnear connectons n T ms M T N. Ths study s n our attenton. References [] Atanasu Gh. New Aspects n Dfferental Geometry of the Second Order Sem. de Mecancă Unv. de Vest dn Tmşoara No [] Atanasu Gh. Lnear Connectons n the Dfferental Geometry of Order Two n the volume: Lagrange and Hamlton Geometres and Ther Applcatons ed. R. Mron Far Partners Publ. Bucureşt [3] Atanasu Gh. Calculus wth N-and JN-lnear connectons on the tangent bundle of the 4 order two Torsons and curvatures (to appear.

13 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS [4] Mron R. The Geometry of Hgher Order Lagrange Spaces Applcatons to Mechancs and Physcs Kluwer Acad. Publ. FTPH no [5] Mron R. and Atanasu Gh. Compendum on the hgher-order Lagrange spaces: The geometry of k-osculator bundle. Prolongaton of the Remannan Fnsleran and Lagrangan structures. Lagrange spaces Tensor N.S [6] Mron R. and Atanasu Gh. Compendum sur les espaces Lagrange d ordre supéreur: La géométre du fbré k-osculateur; Le prolongement des structures Remannennes Fnslerennes et Lagrangennes; Les espaces Lagrange L (kn Sem. de Mecancă Unv. de Vest dn Tmşoara No [7] Mron R. and Atanasu Gh. Lagrange Geometry of Second Order Matt. Comput. Modellng Pergamon 0 4/ [8] Mron R. and Atanasu Gh. Dfferental Geometry of the k-osculator Bundle Rev. Roumane Math. Pures et Appl. 4 3/ [9] Obădeanu V. Structures géométrques assocées à certans systèmes dynamques Sem. de Mecancă Unv. de Vest dn Tmşoara No [0] Olver P.J. Applcatons of Le Groups to Dfferental Equatons Graduate Texts n Mathematcs vol. 07 Sprnger-Verlag New York 986. [] Oprou V. Some remarkable structures and connectons defned on the tangent bundle Rendcont d Matematca 6( [] Purcaru M. On Transformatons Groups of N-Lnear Connectons n the Bundle of Acceleratons General Mathematcs [3] Purcaru M. Remarkable Geometrcal Structures n the Second Order Tangent Bundle (n Romannan Ph.D. Thess Unv. Babeş-Bolya Cluj-Napoca [4] Puta M. Hamltonan Mechancal Systems and Geometrc Quantzaton Kluwer Acad. Publ [5] Vocu N. Devatons of Geodescs n the Geometry of the Order Two (n Romannan Ph.D. Thess Unv. Babeş-Bolya Cluj-Napoca [6] Yano K. and Ishhara S. Tangent and Cotangent Bundles. Dfferental Geometry M. Dekker Inc. New York

14 GHEORGHE ATANASIU AND MONICA PURCARU Department of Algebra and Geometry Translvana Unversty 50 Iulu Manu Street Braşov Româna E-mal address: gh 6