  We obtain an $L^4$ space-time Strichartz inequality for any smooth
three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at
infinity and non-trapping, where $u$ is a solution to the Schr\"odinger
equation $iu_t + {1/2} \Delta_M u = 0$. The exponent $H^{1/4}(M)$ is sharp, by
scaling considerations. In particular our result covers asymptotically flat
non-trapping manifolds. Our argument is based on the interaction Morawetz
inequality introduced by Colliander et al., interpreted here as a positive
commutator inequality for the tensor product $U(t,z',z'') := u(t,z') u(t,z'')$
of the solution with itself. We also use smoothing estimates for Schr\"odinger
solutions including a new one proved here with weight $r^{-1}$ at infinity and
with the gradient term involving only one angular derivative.
