Linear connections induced by a nonlinear connection in the geometry of order two

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1 Linear connections induced by a nonlinear connection in the geometry of order two by Ioan Bucataru 1 and Marcel Roman Pour une variété différentiable, n-dimensionelle M, nous considerons Osc 2 M l éspace total du fibré osculator de ordre deux, sur lequel nous preudrons une connexion nelineaire N. On donnera une connexion lineaire sur Osc 2 M qui depend seulement de N. Cette connexion s appele connexion Berwald. Pour une connexion lineaire arbitraire sur Osc 2 M nous construirons une connexions qui garde des distributions horizontales et verticales. Cette connexion est un extension naturelle de la connexion Vrănceanu de cas du fibre tangent. Est donée la condition dans laqeulle les connexions Berwald et Vrănceanu coincident. MS Classification: 58A20, 58A30, 53B05 Keywords: nonlinear connections, Vrănceanu and Berwald connections 1 Introduction The notion of nonholonomic manifold (distribution, connection) was introduced in 1926 by Gh.Vrănceanu, [?]. This concept was used in the theory of jet bundles by Ch. Ehresman, [?]. Then G.Catz, M.Crampin, M. de Leon, W.Sarlet, J.F.Carinena and others reset it. R.Miron and his school paid a special attention to this concept. A first motivation in order to study this concept is coming from differential equations. It is well known that a second order differential equation (SOE) or a semispray determines a nonlinear connection on the tangent bundle of a manifold. To this nonlinear connection one can associate a linear connection on the tangent bundle, usually called Berwald connection. This connection is very useful in Finsler geometry, [?], [?]. The first author gave in [?] a global expression of it. This linear connection was used by E.Martinez and J.F.Carinena, [?] in study the linearisability of a SOE. The Helmotz conditions for the Inverse Problem 1 Partially supported by Grants of the Romanian Academy 1

2 of the Lagrangian Mechanics have been formulated in a geometrical way by Crampin, [?], using the Berwald connection. Our aim in this paper is to define the Berwald connection in the case of 2-osculator bundle, denoted Osc 2 M. Given a semispray on Osc 2 M (or a three order differential equation) a nonlinear connection it can be obtained. For this nonlinear connection we construct the Berwald connection and we notice it has a special feature: preserves by parallelism the horizontal and the vertical distributions. Some characterizations of such kind of linear connections appear in [?]. They are called d-linear connection. For an arbitrary linear connection on Osc 2 M and a nonlinear connection N, we provide a special d-linear connection, called Vrănceanu connection. A condition in which the Berwald and the Vrănceanu connection are the same is given. As an application we consider a particular 2-semispray and determine the Berwald connection. 2 Preliminaries In this section we recall the construction of the bundle of jets of order 2 for the mapping from a neighborhood of 0 IR to a manifold M usually denoted by J 2 0 (M) or T 2 M. For some historical reasons, R.Miron called it the 2-osculator bundle and denoted it by Osc 2 M. For M, a smooth manifold of dimension n, let C 0,p (IR, M) be the set of germs of smooth mappings f : I IR M, I being an open neighborhood of 0, with f(0) = p. We say that f, g C 0,p (IR, M) are equivalent up to order 2 if there exists a chart (U, ϕ) around p such that (2.1) d h 0(ϕ f) = d h 0(ϕ g), h = 1, 2, where d denotes Frechet differentiation. It can be seen that if (2.1) holds for a chart (U, ϕ), it holds for any other chart (V, ψ) around p. We denote by Osc 2 pf the equivalence class of f C 0,p (IR, M) and set Osc 2 pm = {Osc 2 pf, f C 0,p (IR, M)}. Then we put Osc 2 M = p M Osc 2 pm and define π : Osc 2 M M by π(osc 2 pf) = p. (Osc 2 M, π, M) will be called the 2-osculator bundle of the manifold M. For a local chart (U, ϕ = (x i )) in p M its lifted local chart in u (π) 1 (p) will be denoted by ((π) 1 (U), Φ = (x i, y i, y )). 2

3 For each u = (x, y, y ) Osc 2 M, the natural basis of the tangent space T u E is { x u, i y u, i y u}. Remark that { } span a y n-dimensional vertical distribution V 2 and { y, } span a 2n - dimensional vertical distribution V 1. i y The tensor field: J = y i dxi + y dyi is called the 2-almost tangent structure on Osc 2 M. It has the properties: 1. J is a tensor field of type (1,1) on Osc 2 M, 2. J(V 1 ) = V 2, 3. J 3 = 0, 4. ImJ = KerJ 2 = V 1, ImJ 2 = KerJ = v 2, 5. rank J = 2n, rank J 2 = n. The vector fields Γ 1 = y i y, 2 Γ = y i + 2y, are called y i y the Liouville vector fields and they are globally defined on Osc 2 M. A vector field S χ(e) is called a semispray or a 2-semispray on Osc 2 M if JS = 2 Γ. The local expression of a semispray is: (2.2) S = y i x + i 2y 3Gi y i y, where the functions G i are defined on every domain of local charts. 3 Nonlinear connection The notion of nonlinear connection was introduced by Gh.Vrănceanu as a nonholonomic manifold, [?]. This concept is very useful in the geometry of the total space of a fibred manifold, and it was investigated by R.Miron and his school in [?], [?]. efinition 3.1 A nonlinear connection on the manifold Osc 2 M is a regular distribution N, supplementary to the vertical distribution V 1, i.e. T u E = N(u) V 1 (u), u Osc 2 M. 3

4 Let be N 0 = N, N 1 = J(N 0 ), V 2 = J(N 1 ), J being the 2-almost tangent structure. We get the following direct decomposition (3.1) T u Osc 2 M = N 0 (u) N 1 (u) V 2 (u), u Osc 2 M. A local basis adapted to this decomposition is given by (3.2) { x i, where x = i y = i y i, x N j i i }, (i = 1,..., n) y y i N j i y N j j i y j. y j, The functions Nj i, Nj i are called the coefficients of the nonlinear connection N. If we consider the projectors h, v 1, v 2 determined by (3.1), we can uniquely write: X = hx + v 1 X + v 2 X, X χ(osc 2 M). Of course, the projectors h, v 1, v 2 have the properties: h + v 1 + v 2 = Id, h 2 = h, v 1 v 1 = v 1, v 2 v 2 = v 2, v 1 v 2 = v 2 v 1 = 0, hv α = v α h = 0, (α = 1, 2) The dual basis of the basis (3.2) will be denoted by where {dx i, y i, y }, (i = 1,..., n). y i = dy i + Mj i y = dy + Mj i dx j dy j + M i j dy j. Between the coefficients Nj i, Nj i and dual coefficients Mj i, Mj i connection N we have the following relations of the nonlinear N i j =M i j, Nj i =Mj i M i m M m j. 4

5 In the adapted basis, the 2-almost tangent structure is given by J = y i dxi + y yi. The projectors h, v 1, v 2 can be written in the form h = x i dxi, v 1 = y i yi, v 2 = y y. For a nonlinear connection N, we define the structure (3.3) θ = x i yi + y i y. The structure θ have the following properties: 1. θ 2 J 2 = h, 2. θ J 2 θ = v 1, 3. J 2 θ 2 = v 2, 4. I d = θ 2 J 2 + θ J 2 θ + J 2 θ 2. Proposition 3.1 The structure θ is integrable if and only if the (1,2)-type curvature tensor of the nonlinear connection vanishes, that means that nonlinear connection is integrable, too. Proof. In the adapted basis, the Nijenhuis tensor N θ for the structure θ is expressed by the curvature tensor of N and its derivatives. q.e.d. Theorem 3.1 [?] Let S = y i on Osc 2 M. Then (3.4) Mj i = Gi x i +2y y j, M i j 3Gi be a 2-semispray y i y = Gi y j, are the dual coefficients of a nonlinear connection on Osc 2 M. A different nonlinear connection was derived from S by R.Miron in [?]. 5

6 4 Induced linear connection Let N be a fixed nonlinear connection on Osc 2 M. Consider h, v 1, v 2 the projectors which correspond to the decomposition (3.1) determined by N. efinition 4.1 A linear connection on Osc 2 M is called a d(distinguished)- linear connection if it preserves by parallelism the horizontal and the vertical distributions. Observe that for a d-connection, the projectors h, v 1, v 2 are absolutely parallel with respect to, that is, h = v 1 = v 2 = 0. It is known from Vrănceanu, [?], that on every manifold endowed with a pair of supplementary distributions one can introduce a d-linear connection. We restate that result in the case of 2-osculator bundle, on which we have 3 distributions. Theorem 4.1 Let be an arbitrary linear connection on Osc 2 M. : χ(osc 2 M) χ(osc 2 M) χ(osc 2 M) defined by Then X Y = h hx hy + v 1 v1 Xv 1 Y + v 2 v2 Xv 2 Y + (4.1) h[v 2 X, hy ] + v 1 [hx, v 1 Y ] + v 2 [v 1 X, v 2 Y ]+ θ 2 [v 1 X, J 2 Y ] + (J h)[v 2 X, (θ v 1 )Y ] + (J v 1 )[hx, (θ v 2 )Y ] is a d-linear connection. Proof. By a straightforward computation we can prove that fx Y = f X Y, X fy = X(f)Y +f X Y and since is additive with respect to the both arguments it is a linear connection. We have X hy = h hx hy + h[vx, hy ] = h X Y, so h = 0. In a similar manner we obtain that v 1 = v 2 = 0, so is a d-linear connection on Osc 2 M. q.e.d. The d-linear connection is called the Vrănceanu connection associated to and N. efinition 4.2 A linear connection on Osc 2 M is called a N-linear connection if preserves by parallelism the horizontal distribution N 0 and the k-tangent structure J is absolutely parallel with respect to. 6

7 It is easy to check that a N-linear connection is a d-linear connection, i.e. preserves by parallelism the vertical distribution N 1 and V 2. Some conditions in order that a linear connection be a N-linear connection were given by the second author in [?]. We give here another characterization for a N-linear connection on Osc 2 M using the structure θ. Theorem 4.2 Let be a linear connection on Osc 2 M. Then is a N- linear connection if and only if preserves by parallelism the vertical distribution V 2 and θ is absolutely parallel with respect to. Proof. We must prove that h = o and J = 0 are equivalent with v 2 = 0 and θ = 0. The both sets of conditions are equivalent with that is expressed in the basis { x, i y, } adapted to the distribution i y N 0, N 1 and V 2 as follows (4.2) x j x = F k i ij x, k x j y = F k i ij y, k x j y = F k ij y, k y j y j x i =Ck ij x i =Ck ij x k, y j x k, y j y i =Ck ij y i =Ck ij y k, y j y k, y j y =Ck ij y =Ck ij y k, y k. q.e.d. In the following, we refer to a N-linear connection as to the set of functions Γ(N) = (Fij, k C k ij, C k ij ), called the local coefficients of the N-linear connection. Now, we associate to the nonlinear connection N a N-linear connection, which depends on N only, called the Berwald connection. This N-linear connection appears in local coordinates for k = 1 in the papers [?] of R.Miron and in the pull-back bundle of the tangent bundle in [?]. A global expression of this connection was given by the first author for the case of the tangent bundle in [?]. 7

8 Theorem 4.3 The map : χ(osc 2 M) χ(osc 2 M) χ(osc 2 M) given by: X Y = h[v 2 X, hy ] + v 1 [hx, v 1 Y ] + v 2 [v 1 X, v 2 Y ]+ (4.3) (θ v 1 )[hx, (J h)y ] + (θ v 2 )[v 1 X, (J v 1 )Y ] + J 2 [v 2 X, θ 2 Y ]+ θ 2 [v 1 X, J 2 Y ] + (J h)[v 2 X, (θ v 1 )Y ] + (J v 1 )[hx, (θ v 2 )Y ] is a N-linear connection on Osc 2 M, which depends on the nonlinear connection N only. Proof. All the operators which define are additive, so is additive with respect to the both argument. As h v 1 = h v 2 = v 1 h = v 1 v 2 = v 2 h = v 2 v 1 = 0 we have fx Y = f X Y. Using that θ v 1 J h = h, θ v 2 J v 1 = v 1, J 2 θ 2 = v 2, θ 2 v 2 J 2 = h, J h θ v 1 = v 1 and J v 1 θ v 2 = v 2 we obtain X fy = X(f)Y + f X Y. So, is a linear connection. In order to prove that is a N-linear connection we give the local expression of in the basis (3.2). In this respect we have x i x j = (θ v 1)[ x i, y j ] = N p i y j x p, x i y j = v 1[ x i, y j ] = N p i y j y p, x i y i y i y i y = (J v 1)[ j x, i x = j (θ2 v 2 )[ y, i y = (θ v 2)[ j y, i y = v 2[ j y, i y j ] = y j ] = y j ] = 8 y j ] = N p i y j N p i y j N p i y j N p i y j y p, x p, y p, y p,

9 y y x = h[ j y, x ] = 0, j y = (J h)[ j y, x ] = 0, j y x = J 2 [ j y, x ] = 0. j So, we have the local coefficients of the Berwald connection (4.4) F k ij = N k j y i, Ck ij= N k j y, Ck ij= 0 q.e.d. Proposition 4.1 If the nonlinear connection N is provided by a 2-semispray, i.e. Nj i = Gi then the local coefficients of the Berwald connection are given y j by: F k ij = Cij k = 2 G k Gp y i y j y 2 G k y y, j Ck ij= 0. 2 G k y p y j Theorem 4.4 Let be a linear connection on Osc 2 M, its associated Vrănceanu connection, and be the Berwald connection. Then = if and only if J = 0. Proof. In the basis (3.2) we have x i x = j Γk ji x, k x i y = F k j ji y, k x i y = F k j ji y. k The condition J = 0 is equivalent with Γ k ji = Fji, k where Γ k ji are the first local coefficients of in the natural basis and Fji k are given by (4.4). This means that and coincide on horizontal distribution. In a similar way we prove that and coincide the vertical distributions. q.e.d. 9

10 5 The Riemannian case Let R n = (M, g) be a Riemannian manifold, γ i jk the Christoffel symbols associated to the metric g = g ij. The functions 3G m = 1 2 (γm jk x i + γ m pjγ p ki )yj y k y i + 3γ m ji y j y defined on every domain of chart, are the local coefficients of a 2-semispray S on Osc 2 M. The canonical nonlinear connection determined by S has the first coefficients N i j (x, y ) = Gi y j = γi jk(x)y k. Taking into account the Proposition 4.1, the local coefficients of the Berwald connection are Fji k = γji; k Cji k = 0; Cji k = 0. In this particular case, the local coefficients of the Berwald connection are the same with those determined by R.Miron [?]. Acknowledgements. The authors are indebted to R.Miron and M.Anastasiei for many illuminating discussions. References [1] Abate, M., A characterization of the Chern and Berwald Connections, Houston Journal of Mathematics, 22, 4(1996), [2] Bucataru, I., Sprays and homogeneous connections in the higher order geometry, Studii şi Cercet. Mat., 50, 5-6(1998), [3] Bucataru, I., The Jacobi fields for a spray on the tangent bundle, (to appear). [4] Crampin, M., On the differential geometry of the Euler-Lagrange equations and the inverse problem of the Lagrangian dynamics, J.Phys.A: Math. Gen. bf 14(1981),

11 [5] Ehresman, Ch., Les prolongements d un espace fibré différentiable, Compte Rend. de l Acad. Sci. Paris 240, 1955, [6] Martinez, E., and Carinena, J.F., Geometric characterisation of linearisable second-order differential equations, Math.Proc. Cambridge Phil.Soc., 119(1996), [7] Martinez, E., Carinena, J.F. and Sarlet, W., erivations of differential forms along the tangent bundle projection II, ifferential Geometry and its Appl., 3(1993), [8] Miron, R., The geometry of higher order Lagrange spaces. Applications to Mechanics and Physics, Kluwer Academic Publishers, ordrecht, FTPH no.82, [9] Miron, R., and Anastasiei, M., The geometry of Lagrange spaces; Theory and Applications, Kluwer Academic Publishers, ordrecht, FTPH no.59, [10] Roman, M., Characterisations of the N-linear connection in the higher geometry, to appear in An. Şt. Univ. Al.I.Cuza Iaşi. [11] Vrănceanu, Gh., Sur les espaces non holonome, C.R. Acad. Sci. Paris, 183 (1926), 852. Authors address: Ioan Bucataru, Marcel Roman, Al.I.Cuza University, Iasi Gh.Asachi Technical University, epartment of Mathematics, epartment of Mathematics, 6600 Iaşi, Romania 6600, Iaşi, Romania bucataru@uaic.ro mroman@roman.cccis.ro 11