  We consider the Schr\"odinger equation with no radial assumption on real
hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a
large family of admissible pairs. As a first consequence, we get strong
well-posedness results for NLS. Specifically, for small intial data, we prove
$L^2$ and $H^1$ global well-posedness for any subcritical nonlinearity (in
contrast with the Euclidean case) and with no gauge invariance assumption on
the nonlinearity $F$. On the other hand, if $F$ is gauge invariant, $L^2$
charge is conserved and hence, as in the Euclidean case, it is possible to
extend local $L^2$ solutions to global ones. The corresponding argument in
$H^1$ requires the conservation of energy, which holds under the stronger
condition that $F$ is defocusing. Recall that global well-posedness in the
gauge invariant case was already proved by Banica, Carles & Staffilani, for
small radial $L^2$ data and for large radial $H^1$ data. The second important
application of our global Strichartz estimates is "scattering" for NLS both in
$L^2$ and in $H^1$, with no radial or gauge invariance assumption. Notice that,
in the Euclidean case, this is only possible for the critical power
$\gamma=1+\frac4n$ and can be false for subcritical powers while, on hyperbolic
spaces, global existence and scattering of small $L^2$ solutions holds for all
powers $1<\gamma\le1+\frac4n$. If we restrict to defocusing nonlinearities $F$,
we can extend the $H^1$ scattering results of Banica, Carles & Staffilani to
the nonradial case. Also there is no distinction anymore between short range
and long range nonlinearity : the geometry of hyperbolic spaces makes every
power-like nonlinearity short range.
