A completeness result for time-dependent vector fields and applications

We provide a sufficient condition for the completeness of a time-dependent vector field in R^N, generalizing the well-known left-invariance condition on Lie groups. This result can be applied to the construction of Lie groups associated to suitable families X of Hörmander vector fields, without the need to use the Third Fundamental Theorem of Lie. Further applications are given to the control-theoretic distance related to X, and to the existence of the relevant geodesics.