The adiabatic classical dynamics of certain constrained nonautonomous systems is discussed in terms of the transport of vectors on a surface in parameter space. The classical analogue of Berry’s phase is related to the integrated geodesic curvature of a curve on such a surface and the motion of the system analyzed in terms of the relative rotation rates of different vector fields. The classical dependence of the motion on the topology of the parameter space motivates a particular quantum description of a particle confined to a closed space tube.