Geodesic completeness for Type~A surfaces

Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces are geodesically complete, that some others admit an incomplete geodesic but model geodesically complete surfaces, and that there are also others which do not model any complete surface. Our main result provides a way of determining whether a given set of constant Christoffel symbols can model a complete surface.