Standard paths to connect string theory to cosmology or particle physics require to find backgrounds where the space-time is a product of de Sitter or Minkowski space-time and a compact manifold. We study the existence of such backgrounds at the classical level, in the framework of type II supergravities with parallel orientifolds and D-branes. For de Sitter, we obtain highly constraining no-go theorems; it allows us to exclude a stringy realisation of a particular inflation scenario. For Minkowski, we characterise a broad class of solutions, that possibly encounters for all such backgrounds.
