E. Mart'inez, J. F. Cari~nena Geometric characterization of linearisable second order differential equations. (31K, LaTeX) ABSTRACT. Given an Ehresmann connection on the tangent bundle $\map{\tau}{TM}{M}$ we define a linear connection on the pull--back bundle $\tau^*(TM)$. With the aid of this tool, necessary and sufficient conditions for the existence of local coordinates in which a system of second--order differential equations is linear are derived.