On the $k$-Semispray of Nonlinear Connections in $k$-Tangent Bundle Geometry

In this paper we present a method by which is obtained a sequence of $k$-semisprays and two sequences of nonlinear connections on the $k$-tangent bundle $T^kM$, starting from a given one. Interesting particular cases appear for Lagrange and Finsler spaces of order $k$.


Introduction
Classical Mechanics have been entirely geometrized in terms of symplectic geometry and in this approach there exists certain dynamical vector field on the tangent bundle T M of a manifold M whose integral curves are the solutions of the Euler-Lagrange equations. This vector field is usually called spray or secondorder differential equation (SODE). Sometimes it is called semispray and the term spray is reserved to homogeneous second-order differential equations ( [7], [15]). Let us remember that a SODE on T M is a vector field on T M such that JC = C, where J is the almost tangent structure and C is the canonical Liouville field ( [5], [6]).
In [2], [3], [4] J. Grifone studies the relationship among SODEs, nonlinear connections and the autonomous Lagrangian formalism. In paper [12] Gh. Munteanu and Gh. Pitiş also studied the relation between sprays and nonlinear connectiosn on T M . This study was extended to the non-autonomous case by M. de León and P. Rodgrigues ( [5]). Also, important results for singular non-autonomous case was obtained in [13]. In this paper, following the ideas of papers [10], [11], [12] and [13] we will extend the study of the relationship between sprays and nonlinear connections to the k-tangent bundle of a manifold M . The study of the geometry of this k-tangent bundle was by introduced by R. Miron ([7], [8], [9]). For this case the k-spray represent a system of ordinary differential equations of k + 1 order.

The k-Semispray of a Nonlinear Connection
Let M be a real n-dimensional manifold of class C ∞ and (T k M, π k , M ) the bundle of accelerations of order k. It can be identified with the k-osculator bundle or k-tangent bundle ( [7], [9]).
The following operators in algebra of functions F (T k M ) are k vector fields, globally defined on T k M and linearly independent on the [7]). In applications we shall use also the following nonlinear operator, which is not a vector field, Under a coordinates transformation (1) on T k M , Γ changes as follows: A k-tangent structure J on T k M is defined as usually ( [7]) by the following F (T k M )-linear mapping J : X (T k M ) → X (T k M ): J is a tensor field of type (1, 1), globally defined on T k M .
Obviously, there not always exists a k-semispray, globally defined on T k M . Therefore the notion of local k-semispray is necessary. For example, if M is a paracompact manifold then on T k M there exists local k-semisprays ( [7]).
) i) A k-semispray S can be uniquely written in local coordinates in the form: ii) With respect to (1) the coefficients G i (x, y (1) , ..., y (k) ) change as follows: iii) If the functions G i (x, y (1) , ..., y (k) ) are given on every domain of local chart of T k M , so that (9) holds, then the vector field S from (8) is a k-semispray.
Let us consider a curve c : I → M , represented in a local chart (U, ϕ) by Like in the case of tangent bundle, an Euler Theorem holds. That is, a function Then a k-semispray S is a k-spray if and only if The dimension of horizontal distribution N is n.
If the base manifold M is paracompact then on T k M there exists the nonlinear connections ( [7]).
There exists a unique local basis, adapted to the horizontal distribution N , are called the primal coefficients of the nonlinear connection N and under a coordinates transformation (1) on T k M this coefficients are changing by the rule: Conversely, if on each local chart of T k M a set of functions N i is given so that, according to (1), the equalities (16) hold, then there exists on T k M a unique nonlinear connection N which has as coefficients just the given set of function ( [7]).

The local adapted basis
is given by (15) and and the dual basis (or the adapted cobasis) of adapted basis is . Conversely, if the adapted cobasis δx i , δy (1)i , . . . , δy (k)i i=1,n is given in the form (18), then the adapted basis which is given on each domain of local chart on T k M , so that, according to (1), the relations hold: Let c : I → M be a parametrized curve on the base manifold M , given by x i = x i (t), t ∈ I. If we consider its k-extension c to T k M , then we say that c is an autoparallel curve for the nonlinear connection N if its k-extension c is an horizontal curve, that is d c dt belongs to the horizontal distribution.
From (18) and are characterized by the system of differential equations ( [7]): Now, let be S = 1 S a k-semispray with the coefficients y (1) , ..., y (k) ) like in (8). Then the set of functions gives the dual coefficients of a nonlinear connection N determined only by the k-semispray S (see the book [7] of Radu Miron).
Other result, obtained by Ioan Bucȃtaru ([1]), give a second nonlinear connection N * on T k M determined only by the k-semispray S. That is, the following set of functions is the set of dual coefficients of a nonlinear connection N * .
Let us consider the set of functions ( 2 G i (x, y (1) , ..., y (k) )), given on every domain of local chart by (26) Using (5) Obviously, there exists two nonlinear connections on T k M , which depend only by the k-semispray By this method is obtained a sequence of k-semisprays , ..., ).
We remark that the converse of this proposition is generally not valid and we have the result: We have that g ij is a d-tensor field on the manifold T k M , covariant of order 2, symmetric (see [7]). If (31) rank ||g ij || = n, on T k M we say that L(x, y (1) , ..., y (k) ) is a regular (or nondegenerate) Lagrangian. The existence of the regular Lagrangians of order k is proved for the case of paracompacts manifold M in the book [7] of Radu Miron.  (1) , ..., y (k) ) ∈ T k M → L(x, y (1) , ..., y (k) ) ∈ R, for which the quadratic form Ψ = g ij ξ i ξ j on T k M has a constant signature.
L is called the fundamental function and g ij the fundamental (or metric) tensor field of the space L (k)n .
It is known that for any regular Lagrangian of order k, L(x, y (1) , ..., y (k) ), there exists a k-semispray S L determined only by the Lagrangian L (see [7]). The coefficients of S L are given by This k-semispray S L depending only by L will be called canonical. If L is globally defined on T k M , then S L has the same property on T k M .
From (24) and (25)  Interesting results appear for Finsler spaces of order k. iii) F is k-homogeneous; iv) the Hessian of F 2 with elements (33) g ij = 1 2 The function F is called the fundamental function and the d-tensor field g ij is called fundamental (or metric) tensor field of the Finsler space of order k, F (k)n .
The class of spaces F (k)n is a subclass of spaces L (k)n . Taking into account the k-homogeneity of the fundamental function F and 2k-homogeneity of F 2 we get: 1. the coefficients G i of the canonical k-semispray S F 2 , determined only by the fundamental function F , is (k + 1)-homogeneous functions, that is S F 2 is a k-spray; 2. the dual coefficients of the Cartan nonlinear connection N associated to Finsler space of order k, F (k)n (see [8]), , are homogeneous functions of degree 1, 2, ..., k, respectively, and the primal coefficients has the same property; 3. the dual coefficients of Bucȃtaru's connection N * associated to Lagrangian F 2 are also homogeneous functions of degree 1, 2, ..., k, respectively, and the primal coefficients has the same property.
Using the previous results, we obtain the results:

Conclusions
In this paper was studied the the relation between semisprays and nonlinear connections on the k-tangent bundle T k M of a manifold M . This results was generalized by the author from the 2-tangent bundle T 2 M ( [11]). More that, the relationship between SOPDEs and nonlinear connections on the tangent bundle of k 1 -velocities of a manifold M (i.e. the Whitney sum of k copies of T M , T 1 k M = T M ⊕ · · · ⊕ T M ) was studied by F. Munteanu in [14] (2006) and by N. Roman-Roy, M. Salgado, S. Vilarino in [15] (2011).