  We prove the (local in time) Strichartz estimates (for the full range of
parameters given by the scaling unless the end point) for asymptotically flat
and non trapping perturbations of the flat Laplacian in $\R^n$, $n\geq 2$. The
main point of the proof, namely the dispersion estimate, is obtained in
constructing a parametrix. The main tool for this construction is the use of
the FBI transform.
