Abstract. R. Miron and Gh. Atanasiu studied the geometry of Osc^kM. Among many various problems which was solved, they introduced the adapted basis, the d-connection and gave its curvature theory. Different structures as almost product structure, metric structure was determined and the spray theory was given.
Here the attention on E = Osc³M will be restricted. In Osc³M the Liouville vector fields have important role at definition of almost contact structure J and using J and the Liouville vector fields the sprays are defined. The geodesic lines are integral curves of sprays. The Zermello's conditions which give the independence of the integral of action from the parametrization of the curve are also expressed by the Liouville vector fields.
Almost all the results obtained here can be found in Miron's book [19], [20], even for the space Osc^kM, but here the transformation group is slightly different and the methods of some proofs are new.