  In this paper we develop a quantitative version of Enss' method to establish
global-in-time decay estimates for solutions to Schr\"odinger equations on
manifolds. To simplify the exposition we shall only consider Hamiltonians of
the form $H := - {1/2} \Delta_M$, where $\Delta_M$ is the Laplace-Beltrami
operator on a manifold $M$ which is a smooth compact perturbation of
three-dimensional Euclidean space $\R^3$ which obeys the non-trapping
condition. We establish a global-in-time local smoothing estimate for the
Schr\"odinger equation $u_t = -iHu$. The main novelty here is the
global-in-time aspect of the estimates, which forces a more detailed analysis
on the low and medium frequencies of the evolution than in the local-in-time
theory. In particular, to handle the medium frequencies we require the RAGE
theorem (which reflects the fact that $H$ has no embedded eigenvalues),
together with a quantitative version of Enss' method decomposing the solution
asymptotically into incoming and outgoing components, while to handle the low
frequencies we need a Poincare-type inequality (which reflects the fact that
$H$ has no eigenfunctions or resonances at zero).
