Foliated backgrounds for M-theory compactifications (I)

We summarize our geometric and topological description of compact eight-manifolds which arise as internal spaces in $\mathcal{N}=1$ flux compactifications of M-theory down to ${\mathrm{A}\mathrm{d}\mathrm{S}}_3$, under the assumption that the internal part of the supersymmetry generator is everywhere non-chiral. Specifying such a supersymmetric background is {\em equivalent} with giving a certain codimension one foliation defined by a closed one-form and which carries a leafwise $G_2$ structure, a foliation whose topology and geometry we characterize rigorously.