The jet spaces of maps(Error rendering LaTeX formula) and of the sections of a bundle(Error rendering LaTeX formula) are considered,analysing their vector and affine structures. An intrinsic characterization of second order jet spaces is given. When the bundle η is affine or linear,further results are given. Some interesting maps concerning jet spaces are defined. The k-Lie derivable bundles are introduced and the k-Lie derivative is defined: particular cases are connections(Error rendering LaTeX formula), usual Lie-derivaties(Error rendering LaTeX formula) and Lie-derivaties of geometrical objects. Connections on a bundle are analysed and related with the affine structures of jet spaces and tangent spaces.

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