  We study the nonlinear Schr\"odinger equation posed on product spaces
$\mathbf R^n\times \mathcal M^k$, for $n\geq 1$ and $k\geq1$, with $\mathcal
M^k$ any $k$-dimensional compact Riemaniann manifold. The main results concern
global well-posedness and scattering for small data solutions in non-isotropic
Sobolev fractional spaces. In the particular case of $k=2$, $H^1$-scattering is
also obtained.
