Summary: We study the nature of the nonlinear Schrödinger equation ground states on the product spaces \(\mathbb R^n\times M^k\), where \(M^k\) is a compact Riemannian manifold. We prove that for small \(L^2\) masses the ground states coincide with the corresponding \(\mathbb R^n\) ground states. We also prove that above a critical mass the ground states have nontrivial \(M^k\) dependence. Finally, we address the Cauchy problem issue, which transforms the variational analysis into dynamical stability results.