CONTENTS 0. Introduction...................................................................................................5 1. Natural bundles...........................................................................................10 2. Liftings of functions.....................................................................................15 3. Liftings of functions to the r-frame bundle...................................................22 4. A space of liftings of functions.....................................................................26 5. Quasi-liftings and liftings of vector fields to a natural bundle.......................32 6. Liftings of vector fields to tangent bundles of $p^r$-velocities.....................39 7. Decomposition theorem...............................................................................44 8. Liftings of vector fields to some associated fibre bundles............................49 9. Final remarks...............................................................................................53 References......................................................................................................54