ON THE GROUP OF TRANSFORMATIONS OF SYMMETRIC CONFORMAL METRICAL N-LINEAR CONNECTIONS ON A HAMILTON SPACE OF ORDER K.

Transcript

1 Bulletin of the Transilvania University of Braşov Vol 756 No eries III: Mathematics Informatics Physics ON THE GROUP OF TRANFORMATION OF YMMETRI ONFORMAL METRIAL N-LINEAR ONNETION ON A HAMILTON PAE OF ORDER Monica PURARU 1 Abstract In present paper we obtain in a Hamilton space of order k the transformation laws of the torsion and curvature tensor fields with respect to the transformations of the group T N of the transformations of N linear connections having the same nonlinear connection N. We also determine in a Hamilton space of order k the set of all metrical semisymmetric N linear connections ms T N in the case when the nonlinear connection is fixed and we prove that ms T N together with the composition of mappings s a group. We obtain some important invariants of group ms T N and give their properties. We also study the transformation laws of the torsion d tensor fields with respect to the transformations of the group ms T N Mathematics ubject lassification: 53B05 ey words: Hamilton space of order k nonlinear connection N linear connection metrical N linear connection emetrical semisymmetric N linear connection transformations group subgroup torsion curvature invariants. 1 Introduction The notion of Hamilton space was introduced by R. Miron in [2] [3]. The Hamilton spaces appear as dual via Legendre transformation of the Lagrange spaces. Differential geometry of the dual bundle of k osculator bundle was introduced and studied by R. Miron [5]. In the present section we keep the general setting from R. Miron [5] and subsequently we recall only some needed notions. For more details see [5]. Let M be a real n dimensional manifold and let T k M π k M k 2 k N be the dual bundle of k osculator bundle or k cotangent bundle where the total space is: T k M = T k 1 M T M Transilvania University of Braşov Faculty of Mathematics and Informatics Iuliu Maniu 50 Braşov Romania mpurcaru@unitbv.ro

2 58 Monica Purcaru Let x i y 1i... y k 1i p i i = n be the local coordinates of a point u = x y 1... y k 1 p T k M in a local chart on T k M. The change of coordinates on the manifold T k M is: x i = x i x 1... x n det x i 0 x j ỹ 1i = xi y 1j x j... k 1 ỹ k 1i = ỹk 2i y 1j k 1 ỹk 2i y k 1j x j y k 2j p i = xj p x i j. 1.2 We denote by T k M = T k M \ 0 where 0 : M T k M is the null section of the projection π k. Let us consider the tangent bundle of the differentiable manifold T k M : T T k M dπ k T k M where dπ k is the canonical projection and the vertical distribution V : u T k M V u T u T k M locally generated by the vector y k 1i fields:... at every point u T k M. y 1i The following F T k M linear mapping: J : χ T k M χ T k M defined by: J p i x i = y 1i J = y k 1i J y k 1i y 1i =... J y 2i y k 2i = = 0 J = 0 u T p k M 1.3 i is a tangent structure on T k M. We denote with N a nonlinear connection on the manifold T k M with the coefficients: j N i 1 x y 1... y k 1 p j... N i x y 1... y k 1 p N ij x y 1... y k 1 p k 1 i j = n. The tangent space of T k M in the point u T k M is given by the direct sum of vector spaces: T u T k M = N 0u N 1u... N k 2u V k 1u W ku u T k M 1.4 A local adapted basis to the direct decomposition 1.4 is given by: x i... y 1i y k 1i i = n 1.5 p i

3 On the group of transformations 59 where: = x i N j x i i... 1 y 1j N k 1 j i = y N j 1i y 1 i i... N j 1 y 2j i k 2... = y k 1i y k 1i p i = p i. y k 1j + N ij y k 1j p j 1.6 Under a change of local coordinates on T k M the vector fields of the adapted basis transform by the rule: x i = xj x i x j xj = y 1i x i... ỹ 1j xj = y k 1i x i The dual basis of the adapted basis 1.5 is given by: where: ỹ k 1j = xi p i x j p j. 1.7 x i y 1i... y k 1i p i 1.8 dx i = x i dy 1i = y 1i N i j x j 1... dy k 1i = y k 1i N i j y k 2j... N i j y 1j N i j x j 1 k 2 k 1 dp i = p i + N ji x j. 1.9 With respect to 1.2 the covector fields 1.8 are transformed by the rules: x i = xi x j ỹ 1i = xi y 1j... ỹ k 1i = xi y k 1j x j x j x j p i = xj p x i j Let D be an N linear connection on T k M with the local coefficients in the adapted basis 1.5 : DΓ N = H i i jh jh jh i α = 1... k An N linear connection D is uniquely represented in the adapted basis in the following form:

4 60 Monica Purcaru D x j D x j D D = H s x i ij x D s p i = Hsj i p s = y j y j D p j D p j x i p i s ij = i sj x j x s D x i = i js x s D p j p i = ij s p s. y i y j = H s ij α = 1... k 1 y s y βi s ij y βs = p s α β = 1... k 1 y i = i js y s α = 1... k The transformations of the d tensors of torsion and curvature In the following we shall study the Abelian group T N. Its elements are the transformations t : DΓ N D Γ N given by see2.11-p.131 [6]: i i N j = N j α = 1... k 1 N ij = N ij H i jh = H i jh B i jh i i i jh = jh D jh α = 1... k 1 jh i = jh i D jh i i j h = n. 2.1 Firstly we shall study the transformations of the d tensors of torsion of DΓ N. Proposition 2.1. The transformations of the Abelian group T N given by 2.1 lead to the transformations of the d tensors of torsion in the following way: r i jh = r i i i jh B jh = B jh B ijh = B ijh B h h ij = B ij B ijh = B ijh B i jh R 0k 1 i jh = = B jh i R i jh = R i jh R i jh = R i jh R i jh R ijh = R ijh α β α β = 1... k 1 0k T i jh = T i jh + B i hj B i jh i i i i jh = jh + D hj D jh jh i = jh i D hj jh i D i 2.4 γ i jh = γ i jh α β α β γ = 1... k 1 i jh α1 = i jh α1... i jh αk 1 = i jh αk 1 α = 1... k

5 On the group of transformations 61 Proof. Using p.131 [5] 5.12 p.135 [5] 6.3 p.161 [5] and 2.1 we have the results. Now we shall study the transformations of the d tensors of curvature of DΓ N. We get: Proposition 2.2. The transformations of the Abelian group T N given by 2.1 lead to the transformations of the d tensors of curvature in the following way: R i m jh = R i m jh D i ms R s jh... D i ms R s jh D is m R sjh 1 01 k 1 0k 1 0 B i mst s jh + A jh B s mj B i sh B i 2.6 mj h P mi jh = P m i jh B i mj h + D i mh j B i ms s jh + D i ms Hhj s + Bs mj D i sh D s mh B i sj D i ms B s jh... 1 α1 D B i ms k 1 αk 1 α = 1... k 1 P m i j h = P m i j h B i mj h +D ih m j Bi ms sh j D sh m B i sj D 1i ms B 1 j sh... D k 1 i ms s jh D is m B sjh D is mh h sj + Bs mj Dih s B j sh Dm is B h sj k 1 0 mi jh = m i β jh D i mj h + D i mh j D i ms s jh + D i ms s jh + β β β +D s mj D β i sh D β s mh D i sj D 1 i ms 1 mi j h = m i j h D i mj h + D ih m s jh... D is m B sjh α β α β = 1... k 1. j sh j Dm sh D i sj D i ms j sh... D i ms 1 α1 k 1 αk 1 jsh Dm is α = 1... k 1 k 1 D i ms s jh k D i ms h sj D is m + D s mj D ih s h sj 2.10 m ijh = m ijh + D m is s jh + A jh D m ij h +D m sj D s ih 2.11 where A ij denotes the alternate summation and m m and m denote the h- covariant derivative the v α -covariant derivative and the w k -covariant derivative with respect to DΓN respectively α = 1... k 1. Proof. Using 6.4-p.161 [5] 2.1 and 5.2 -p.156 [5] we have the results. We shall consider the tensor fields: i m jh = R i m jh i 1 ms R s 01 jh... i k 1 ms R s 0k 1 jh is m R 0 sjh 2.12

6 62 Monica Purcaru P mi jh = A jh P m i jh i 1 ms B s α1 jh... i α = 1... k 1 k 1 ms B s αk 1 jh is m B sjh 2.13 P mi jh = P m i jh i 1 ms B s α1 jh... i k 1 ms B s αk 1 jh m is B sjh 2.14 α = 1... k 1 i P h i m j = A jh P h m j i 1 ms B sh 1 j... i k 1 ms B sh k 1 j mb is h 0 sj 2.15 P m i j h = P m i j h i 1 ms B sh 1 j... i k 1 ms B sh k 1 j is mb h 0 sj 2.16 mi jh = A jh mi jh = m i jh i m i j h = A jh mi jh i 1 1 ms 1 1 ms s jh i k 1 ms k 1 α β α β = 1... k 1 s jh... i k 1 ms k 1 α β α β = 1... k 1 i h m j i 1 ms sh α1 j... i α = 1... k 1 mi j h = m i j h i 1 ms sh α1 j... i s jh is m B sjh s jh is m B sjh k 1 ms sh αk 1 j m is h sj k 1 ms sh αk 1 j m is h sj α = 1... k Proposition 2.3. By a transformation of the Abelian group T N given by 2.1 the tensor fields m i jh P mi jh P mi jh P i h m j P m i j h are transformed according to the following laws: m i jh mi jh m i j h mi j h i i m jh = m jh B i mst s jh + A jh B s mjb i sh B i mj h 2.21 P mi jh = P m i jh + D i ms T s jh Bi ms +B s mj D i sh D s mh B i sj B i mj s jh + A jh α = 1... k 1 h + D i mh j P mi jh = P mi jh Bmj i h + D i mh j B i ms s jh D i ms Hjh s B s mj D i sh D s mh Bsj i α = 1... k 1

7 On the group of transformations 63 P i h m j = P i h m j + A jh Bmj i h + D ih m j B i msj sh DmH is sj h + +Bmj s Dih s Dm sh Bsj i α = 1... k P m i j h = P m i j h B i mj h + D m ih j B i ms sh j m i jh = i m jh D i ms s β jh D i +D s mj D i β sh D s β mh D i sj β ms s D is mh h sj + Bs mj Dih s D i mj jh + A jh α β α β = 1... k 1 D sh m B i sj β h + D i β mh j m i jh = i m jh D i mj β h + D i β mh j D i ms s β jh + D i β ms s hj + +D s mj D i β sh D s β mh D i sj α β α β = 1... k m i j h = m i j h + A jh +D s mj Ds ih D i mj h + Dm ih D sh m D i sj j j shdi ms h sj Dm+ is α = 1... k i m h j = i m h j D i mj h + Dm ih j j shdi ms h sj Dm+ is +D s mj Ds ih Dm sh D i sj α = 1... k Proof. From 2.7 we get: i i A jh P m jh = A jh P m jh + A jh +A jh B i mj D i ms H s hj Bi ms s jh + i h + D mh j + B s i s mj D sh D mh B i i s sj D ms B jh... 1 α1 s D B jh Dm is B sjh i ms k 1 αk 1

8 64 Monica Purcaru Using p.161[5] and 2.1 we have: i i i A jh P m jh = A jh P m jh D ms Tjh s Bi ms +A jh B i mj s jh + h + D i mh j + B s mj D i sh D s mh B i sj + +A jh i i s ms ms B jh α1 + m is m is B sjh. i i s ms ms B jh + k 1 k 1 αk 1 If we separate the terms and using 2.13 we get Analogous we obtain the other formulas. 3 Metrical semisymmetric N linear connections of the space H kn Let H kn = M H be a Hamilton space of order k and let N be the canonical nonlinear connection of space H kn = M H [5] p.192. x i We consider the adapted basis... and its dual basis y 1i x i y 1i... y k 1i p i determined by N and by the distribution Wk. Let y k 1i p i g ij x y 1... y k 1 p = 1 2 H p i p j be the fundamental tensor of the space H kn. The d-tensor field g ij being nonsingular on T k M there is a d-tensor field g ij covariant of order 2 symmetric uniquely determined at every point u T k M by g ij g jk = k i. 3.2 Definition 3.1. [5] An N linear connection D is called compatible to the fundamental tensor g ij of the Hamiltonian space of order k H kn = M H or it is metrical if g ij is covariant constant or absolutelyy parallel with respect to D i.e. g ij ij h = 0 g h = 0 g ij h = 0 α = 1... k The tensorial equations 3.3 imply: g ij h = 0 ij g h = 0 g ij h = 0 α = 1... k

9 On the group of transformations 65 Theorem 3.1. [5] 1. In a Hamilton space of order k H kn = M H there is a unique N-linear connection D with the coefficients DΓ N = H i i i jh jh... jh jh i verifying the axioms: 1 k 1 1. N is the canonical nonlinear connection of H kn. 2. The fundamental tensor g ij is h v α and w k - covariant constant: g ij h = 0 gij h = 0 g ij h = DΓN is h v α and w k - torsion free T i jh = i jh = i jh = 0. α = 1... k 1 2. The connection DΓN has the coefficients given by the generalized hristoffel symbols: H i jh = 1 2 gis gsh x j i jh = 1 2 gis gsh i jh = 1 2 g is + g js x h g jh x s + g js g jh y j y h y s g sh p j + gjs p h gjh p s α = 1... k This connection depends only on the fundamental function H of the space H kn and on the canonical nonlinear connection N. Definition 3.2. [5] The connection DΓN with the coefficients 3.5 will be denoted by ΓN and called canonical for the space H kn. The canonical metrical N-linear connection: ΓN has zero torsions Tjh i i jh jh i α = 1... k 1. The Obata s operators are given by: There is inferred: Ω ij hk = 1 h i 2 j k g hkg ij Ω ij hk = 1 h i 2 j k + g hkg ij. 3.6 Proposition 3.1. The Obata s operators have the following properties: Ω ir sj + Ω ir sj = i s r j 3.7 Ω ir sjω sn mr = Ω in mj Ω ir sjω sn mr = Ω in mj Ω ir sjω sn mr = Ω ir sjω sn mr = Ω ir rj = Ω ir si = 0 Ω ir ij = 1 2 n 1 r j Ω ir ij = 1 2 n + 1 r j. 3.9 Theorem 3.2. [4] [5] There is a unique metrical connection DΓ N = H i jh i jh jh i α = 1... k 1 metrical with respect to the fundamental tensor g ij of space H kn having as torsion d tensor fields T i jh i jh i jh apriori given.

10 66 Monica Purcaru The coefficients of DΓ N have the following expressions: H i jh = 1 2 gis gsh + g js g jh x j x h x + s gis g sm T m jh g jm T m sh + g hm T m js i jh = 1 2 gis gsh + g js g jh + y j y h y s g gis sm m jh g jm m sh + g hm m jh i = 1 2 g g sh is p j + gjs p h gjh p s g sm jh m g jm sh m + g hm js m 1 2 g is js A class of metrical N-linear connections which has interesting properties is that of metrical semisymmetric N-linear connection. Definition 3.3. [1] An N linear connection on T k M is called semisymmetric if: T i jh = 1 i 2 j σ h + h i σ j i jh = 1 j i τ h + h i 2 τ j jh i = 1 j i 2 vh + i h v j α = 1... k where σ τ χ T k M and v χ T k M. Theorem 3.3. The set of all metrical semisymmetric N linear connections with local coefficients DΓ N = H i i i jh jh... jh jh i is given by: 1 k 1 gjh g is σ s + σ j h i H i jh = 0 H i jh i jh = 0 where Γ N = i jh i jh = 0 i jh g jh g is τ s + τ g jh g is v s + v j i h 0H i jh 0 i jh k 1 α = 1... k j i h i jh 0 jh i are the local coefficients of the canonical metrical N linear connection and σ τ χ T k M α = 1... k 1 and v χ T k M. Proof. Using Theorem 3.2 and Definition 3.3 we obtain the results by direct calculation. 4 The group of transformations of metrical semisymmetric N linear connections Let N be a given nonlinear connection on T k M.

11 On the group of transformations 67 Remark 4.1. The relations 3.12 give the transformations of metrical semisymmetric N linear connections of space H kn corresponding to the same nonlinear connection N. We shall denote a transformation of this form by: tσ τ v : DΓN DΓN α = 1... k 1. It is given by: H i jh = H i jh gjh g is σ s + σ j i h i jh = i jh g jh g is τ s + τ jh i α = 1... k jh i = jh i g jh g is v s + v j i h. Thus we have: Theorem 4.1. The set ms T N of all transformations tσ τ v : DΓN DΓN α = 1... k 1 of metrical semisymmetric N linear connections given by 4.1 is an Abelian group together with the mapping product. This group acts on the set of all metrical semisymmetric N linear connections corresponding to the same nonlinear connection N transitively. Theorem 4.2. By means of a transformation 4.1 the tensor fields: i m jh P i m jh α = 1... k 1 P i h m j i m jh α β α β = 1... k 1 ijh m given in and 2.11 are changed by the laws: i i m jh = m jh + A jh Ω ir jm σ rh 4.2 P m i jh = P m i jh + A jh Ω ir jm γ rh α = 1... k i P h i m j = P h m j + A jh Ω ir jmσ r h +Ω ij rmv r h + Ω is rm H h sjv r + rh j σ s Ωij hm σ rv r h mg jr g is σ s v r i h g rmg js σ s v r 1 4 g jsg ih σ m v s 1 4 g jmg rh σ r v i m i i jh = m jh + A jh Ω ir τ jm rh α β α β = 1... k where: γ m ijh = h ijh + A jh Ω ij rmv rh 4.6 σ rh = σ r σ h + σ r h g rh σ σ = g rs σ r σ s 4.7 rh = σ r τ h + σ h τ r γ = g σ rs r τ s + σ s τ r + σ r h + τ r h g rh γ α = 1... k 1 4.8

12 68 Monica Purcaru τ rh = τ τ = grs r τ β h + τ h τ r β τ r τ β s + τ s τ β r β + τ r h + τ r h g rh τ β α β α β = 1... k v rh = v r v h + v r h 1 4 grh v v = g rs v r v s Proof. Using 2.1 and 4.1 we get: B i jh = 1 σj h i 2 + g jhg is σ s = Ω is i hj σ s D jh = 1 τ jh i 2 + g jhg is τ s = 4.11 Ω is hj τ s α = 1... k 1 D jh i = 1 v j i h + g jh g is v s = Ω jh si 2 vs. By applying Proposition 2.3 relations 3.11 and 4.11 we obtain the results. Using these results we can determine some invariants of group ms T N. To this end we eliminate σ ij γ ij τ ij and v ij from and 4.6 and we obtain: Theorem 4.3. For n > 2 the following tensor fields: H m i jh N m i jh α = 1... k 1 M i m jh M ijh m are invariants of group ms N m i jh = P m i jh + 2 T N : i i H m jh = m jh + 2 n 2 A jh Ω ir jm n 2 A jh Ωir jm P rh rh g rh n 1 g rh P α = 1 k n 1 M m i i jh = m jh + 2 n 2 A jh Ωir jm rh g rh α β; α β = 1 k 1 2 n M ijh m = ijh m + 2 n 2 A jh Ω ij rm rh grh n 1 where: mj = i m ji = g mj mj P mj = P i m ji P = g mj P mj α = 1 k 1 mj = i m ji = g mj mj α β α β = 1 k 1 ij = ijm m = g ij ij.

13 On the group of transformations 69 In order to find other invariants of group ms T N let us consider the transformation formulas of the torsion d tensor fields by a transformation tσ τ v : DΓN DΓN α = 1... k 1 of metrical semisymmetric N linear connections of the space H kn corresponding to the same nonlinear connection N given by 4.1. Proposition 4.1. By a transformation 4.1 of metrical semisymmetric N linear connections corresponding to the same nonlinear connection N : tσ τ v : DΓN DΓN α = 1... k 1 the torsion tensor fields: r i i h jh B jh B ijh B ij B 01 0 ijh B jh i R 0α i jh T i jh R 0 ijh i jh i jh γ i jh α1 ijh... αk 1 ijh α β α β γ = 1... k 1 are transformed as follows: r i jh = r i i i jh B jh = B jh B h ij = B h ij B ijh = B ijh 0α 0α 0 0 B ijh = B ijh B jh i = B jh i α β α β = 1... k 1 i i R jh = R jh α = 1... k 1 R ijh = R ijh 0α 0α γ i jh = γ i jh α β α β = 1... k 1 α1 ijh = i jh... α1 αk 1 ijh = αk 1 ijh α = 1... k 1 T i jh = T i jh A jh σj i h i i jh = jh A jh τ jh i = jh i A jh v j i h i i h 4.16 We denote with: t i jh = A jh N i j t i jh = A jh y βh N jh p i t ih j α β = 1... k 1 N i j = A jh p h α = 1... k

14 70 Monica Purcaru t ijh = Σ ijh g im t t ijh = Σ ijh g im t m jh r ijh = Σ ijh g im r 0α 0α T ijh = Σ ijh g im T m jh m jh t h ij = Σ ijh g im t jmh α β α β = 1... k 1 m jh R ijh = Σ ijh g im R m jh 0α 0α ijh = Σ ijh γ α = 1... k 1 α = 1... k 1 R i jh = Σ ijh g im R mjh γ gim m jh α β α β = 1... k h ij = Σ ijh g im mh j α1 α1 B 1 B i ijh = Σ ijh g im B m jh jh = Σ ijh g im B mjh m g im B jh 0 B ijh = Σ ijh 0 2 B... h ij = Σ ijh g im mh j αk 1 αk 1 α β α β = 1 k 1 α = 1 k 1 B i jh = Σ ijh g im B mjh α = 1 k 1 α β α β = 1 k 1 h ij = Σ ijh g im B mh j α = 1... k 1 m ijh = Σ ijh g im jh α = 1 k 1 ijh = Σ ijh g im jh m H ijh = Σ ijh A jh g jm H m jh ijh = Σ ijh A jh g jm m ih α = 1 k 1 ijh = Σ ijh A jh g jm ih m where Σ ijh... denotes the cyclic summation and with:

15 On the group of transformations 71 1 ijh = g im T m jh A jh g hm H m ij 2 ijh = g im m m jh A jh g hm ij α = 1... k 1 2 γ γ ijh m = A jh g hm ij 3 γ ijh = g mj γ m γ m ih + g im jh α β α β γ = 1... k ijh = g im m jh A jh g hm m ij 1 ijh m = A ij g im B jh 2 ijh m = A jh g mj A ih B ih α β α β γ = 1... k 1. Remark 4.2. It is noted that are alternate 1 ijh 2 t 2 γ ijh ijh 3 ijh t h ij t ijh r ijh T ijh Rjh i ijh ijh 0α ijh 2 ijh H ijh ijh ijh are alternate with respect to: j h and 1 ijh are alternate with respect to i j. Theorem 4.4. The tensor fields: i R jh R ijh γ i jh 0α α1 i jh... r i jh 0α αk 1 ijh r ijh T γ ijh R ijh ijh Rjh i 0α 0α α1 B i jh 2 B h ij ijh ijh 1 ijh 2 ijh are invariants of the group ms T N. i h B jh B ijh B ij 0 t i jh t i jh t ih j h ij... 2 γ ijh ijh h ij B αk 1 3 γ ijh 3 B ijh B jh i t ijh t h ij t ijh ijh 1 B i ijh 1 ijh 2 jh B 0 ijh Proof. By means of transformations of the torsion given in 4.16 and using the notations: and 4.19 from 4.1 we obtain the results by direct calculation.

16 72 Monica Purcaru Theorem 4.5. Between the invariants in Theorem 4.4 there exist the following relations: 1 Σ ijh ijh = T ijh + H 2 ijh = 0 Σ ijh ijh = ijh + ijh = 0 2 γ Σ ijh ijh = γ ijh γ Σ ijh 3 ijh = ijh + ijh = 0 Σ ijh ihj α β α β γ = 1... k 1 3 γ ijh = γ ijh + γ ihj α β α β γ = 1... k 1 1 Σ ijh ijh = B ijh B jih α β α β γ = 1... k 1 2 Σ ijh = A ih jih A ij B hij α β α β = 1... k 1 Σ ij ijh 3 γ B = 0 α β α β γ = 1... k Proof. Using notations relations 4.1 and the definitions of the torsion d tensor fields given in [5] - p.161 we obtain the results. References [1] Atanasiu Gh. and Târnoveanu M. New aspects in the differential geometry of the second order cotangent bundle Univ. de Vest din Timişoara [2] Miron R. ur la geometrie des espaces Hamilton.R. Acad. ci. Paris er II no [3] Miron R. Hamilton Geometry Analele Şt. Univ. Iaşi s. I Mat [4] Miron R. Hrimiuc D. himada H. and abău V.. The geometry of Hamilton and Lagrange spaces luwer Academic Publisher FTPH [5] Miron R.The Geometry of Higher- Order Hamilton paces. Applications to Hamiltonian Mechanics luwer Acad. Publ. FTPH [6] Purcaru M. and Târnoveanu M. On the transformations groups of N-linear connections on teh dual bundle of k-tangent bundle. tudia Univ. Babeş Bolyai Math no