  Some nilpotent Lie groups possess a transformation group analogous to the
similarity group acting on the Euclidean space. We call such a pair a nilpotent
similarity structure. It is notably the case for all Carnot groups and their
dilatations. We generalize a theorem of Fried: closed manifolds with a
nilpotent similarity structure are either complete or radiant and, in the
latter case, complete for the structure of the space deprived of a point. The
proof relies on a generalization of convexity arguments in a setting where, in
the coordinates given by the Lie algebra, we study geodesic segments instead of
linear segments. We show classic consequences for closed manifolds with a
geometry modeled on the boundary of a rank one symmetric space.
