  We obtain the Strichartz inequalities $$ \| u \|_{L^q_t L^r_x([0,1] \times
M)} \leq C \| u(0) \|_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian
manifold $M$ which is asymptotically conic at infinity (with either short-range
or long-range metric perturbation) and non-trapping, where $u$ is a solution to
the Schr\"odinger equation $iu_t + {1/2} \Delta_M u = 0$, and $2 < q, r \leq
\infty$ are admissible Strichartz exponents ($\frac{2}{q} + \frac{n}{r} =
\frac{n}{2}$). This corresponds with the estimates available for Euclidean
space (except for the endpoint $(q,r) = (2, \frac{2n}{n-2})$ when $n > 2$).
These estimates imply existence theorems for semi-linear Schr\"odinger
equations on $M$, by adapting arguments from Cazenave and Weissler \cite{cwI}
and Kato \cite{kato}.
  This result improves on our previous result in \cite{HTW}, which was an
$L^4_{t,x}$ Strichartz estimate in three dimensions. It is closely related to
the results of Staffilani-Tataru, Burq, Tataru, and Robbiano-Zuily, who
consider the case of asymptotically flat manifolds.
