On the Geometries of Superior Order

The geometries of higher order, defined in the present paper as the study of the category of jet bundles (J o M, π,M), were suggested by the old problem of the prolongations to J o M of Riemannian structures g apriori given on the base manifold M. In the case k = 1 there are very good examples: the geometry of Finsler spaces and the geometry of Lagrange spaces, [3]. For k > 1 there are some geometrical results concerning the analytical Mechanics of superior order and there are some important studies on the total space of jet bundles. Recently, together with Gh.Atanasiu, we have considered the notion of k-osculator bundle (OscM,π, M), which can be identified to the bundle (J 0 M, π, M) and, for it, we have studied: a. The Geometry of the total space OscM : vertical distributions, Liouville vector fields, k-tangent structure, sprays, nonlinear connections, N-linear connections, curvatures, torsions, structure equations etc. b. Taking this oportunity we solved the problem of the prolongation to OscM of the Riemannian structure g apriori given on M . The same problem for Finsler and Lagrange structures can be formulated. c. We have defined the notion of higher order Lagrange spaces and studied for them the most important geometrical object fields. d. Recently, to the Tsagas’ workshop, in December 1995, at the ”Aristoteles” University of Thessaloniki, the author presented the theory of subspaces in the higher Lagrange spaces. In the present paper I shall describe, in synthesis, the construction of these geometries. 1. Let M be a real n-dimensional C∞-manifold and (OscM, π, M) its kosculator bundle. We briefly describe the manifold E = OscM . Two curves in M , σ, ρ : I → M , with x0 ∈ M as common point, x0 = σ(0) = ρ(0), 0 ∈ I, have at x0 a ”contact of order k”, (k ∈ N), if for any function f ∈ F(U), x0 ∈ U , (U being an open set in M), satisfy