Geometric structures and configurations of flags in orbits of real forms

Abstract

This is an introduction and a survey on geometric structures modelled on closed orbits of real forms acting on spaces of flags. We focus on 3-manifolds and the flag space of all pairs of a point and a line containing it in \({\mathbb{P}}({\mathbb{C}}^3)\). It includes a description of general flag structures which are not necessarily flat and a combinatorial description of flat structures through configurations of flags in closed orbits of real forms. We also review volume and Chern–Simons invariants for those structures.