  We obtain improved Strichartz estimates for solutions of the Schr\"odinger
equation on negatively curved compact manifolds which improve the classical
universal results results of Burq, G\'erard and Tzvetkov [11] in this geometry.
In the case where the spatial manifold is a hyperbolic surface we are able to
obtain no-loss $L^{q_c}_{t,x}$-estimates on intervals of length $\log
\lambda\cdot \lambda^{-1} $ for initial data whose frequencies are comparable
to $\lambda$, which, given the role of the Ehrenfest time, is the natural
analog of the universal results in [11]. We are also obtain improved endpoint
Strichartz estimates for manifolds of nonpositive curvature, which cannot hold
for spheres.
