  We prove global well-posedness and scattering in $H^1$ for the defocusing
nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases}
&(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases}
\end{equation*} on the hyperbolic spaces $\H^d$, $d\geq 2$, for exponents
$\sigma\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linear
solutions in the case of small exponents $\sigma$; for comparison, on Euclidean
spaces scattering in $H^1$ is not known for any exponent $\sigma\in(1/d,2/d]$
and is known to fail for $\sigma\in(0,1/d]$. Our main ingredients are certain
noneuclidean global in time Strichartz estimates and noneuclidean Morawetz
inequalities.
