On projective models of Thurston geometries, some relevant notes on Nil orbifolds and manifolds

A unified method is given for describing $3$-orbifolds and $3$-manifolds in terms of a projective interpretation of Thurston geometries. As an application, a new characterization is given for Nil-orbifolds and manifolds arising as fiber spaces over Euclidean $2$-orbifolds with prescribed fundamental groups. (Theorems 3.1–3.2, see also Figure 1).