Classical Notions and Problems in Thurston Geometries

TL;DR

This paper surveys classical geometric concepts like geodesics and triangles in Thurston geometries beyond constant curvature, highlighting their structure and posing open questions.

Contribution

It introduces classical geometric notions in non-constant curvature Thurston geometries, which have been less studied, and summarizes recent results and open problems.

Findings

Formulation of classical geometric concepts in non-constant curvature geometries

Summary of recent results on geodesics, triangles, and lattices in these geometries

Identification of open questions for future research

Abstract

Of the Thurston geometries, those with constant curvature geometries (Euclidean $\EUC$, hyperbolic $\HYP$, spherical $\SPH$) have been extensively studied, but the other five geometries, $\HXR$, $\SXR$, $\NIL$, $\SLR$, $\SOL$ 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 -- 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.