  We study small data scattering of solutions to Nonlinear Klein-Gordon
equations with suitable pure power nonlinearities, posed on $\mathbb{R}^d\times
\mathcal{M}^k$ with $k\leq2$ and $d\geq1$ and $\mathcal{M}^k$ a compact
Riemannian manifold. As a special case we cover the $H^1-$critical NLKG on
$\mathbb{R} \times M^2$.
