Locally homogeneous geometric manifolds

TL;DR

This paper surveys the classification of locally homogeneous geometric structures on manifolds, highlighting their role in understanding surface group representations and deformation spaces in low-dimensional topology.

Contribution

It provides an overview of the classification of geometric structures on manifolds and explores their applications in studying surface group representations.

Findings

Classification of locally homogeneous structures in low dimensions

Connection between geometric structures and surface group representations

Insights into deformation spaces of fundamental group representations

Abstract

Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology S and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of G/H into S. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on S, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of S. In this talk I will survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.