A Topological Analogue of Mostow's Rigidity Theorem

Three types of manifolds, spherical, fiat, and hyperbolic, are paradigms of geometric behavior. These are the Riemannian manifolds of constant positive, zero, and negative sectional curvatures, respectively. (The positive and negative sectional curvatures may be assumed, after scaling, to be + 1 and -1.) There is an equivalent synthetic geometric definition of them in terms of coordinate charts for a manifold M. Namely, a spherical, flat, or hyperbolic structure on M is an atlas of coordinate charts {(Un' .t;,)lo: E J} for M, where each .t;, ( Un) is an open subset of the (unit) sphere Sn , Euclidean space R n , or (real) hyperbolic space H n , respectively, such that