Rigidity of geometric structures

Geometric structures on a manifold M arise from a covering of M by charts with values in a homogeneous space G/H, with chart transitions restrictions of elements of G. If M is aspherical, then such geometric structures are given by a homomorphism of the fundamental group of M into G. Rigidity of such structures means that the conjugacy class of the homomorphism can be reconstructed from topological or geometric information on M. We give an overview of such rigidity results, focusing on topological type and length functions.