Classical Notions and Problems in Thurston Geometries

Abstract

Of the Thurston geometries, those with constant curvature geometries (Euclidean $ E^3$, hyperbolic $ H^3$, spherical $ S^3$) have been extensively studied, but the other five geometries, $ H^2\times R$, $ S^2\times R$, $Nil$, $\widetilde{SL_2 R}$, $Sol$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these geometries -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries -- can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.