  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.
