The geodesic flow on the unit tangent bundle of a closed surface of constant negative curvature is one of the earliest examples of an ergodic dynamical system. This was first proven by G. A. Hedlund in 1936. Soon afterwards, it was reproved by E. Hopf, who also generalized to the case of closed surfaces of variable negative curvature. Hopf's proof already indicated the relevance of negative curvature to the ergodicity of the geodesic flow. About 20 years later, Hopf's theorem was generalized by Anosov to geodesic flows on unit tangent bundles of higher dimensional closed negatively curved manifolds.
Around the same time, combining certain basic results in geometry and topology, it was observed that the fundamental group of a closed manifold M of negative curvature determines M up to homotopy equivalence. Mostow's celebrated strong rigidity theorem (1968) showed that, within the class of closed locally symmetric spaces M of non-compact type of dimension ≥ 3, the fundamental group π1(M) determines M up to isometry (possibly after scaling the metric by a positive constant). Further study of this rigidity phenomenon led to two important generalizations. On the one hand, Margulis established his super-rigidity theorem in higher rank. Using the harmonic map techniques of Eells-Sampson, versions of the super-rigidity theorem were established for certain rank 1 cases by Corlette, Jost-Yau and Mok-Siu-Yeung. On the other hand, it led Ballmann-Brin-Eberlein to introduce, in the mid 1980's, the notion of geometric rank. Generalizing a notion that existed for locally symmetric spaces, this culminated in the rank rigidity theorem due, independently, to Ballmann and Burns-Spazier. Around this time, Gromov realized that many results of this nature could be formulated and proved, in a synthetic way, in the more general setting of metric spaces of non-positive curvature. The study of this broader class of spaces resulted in certain new rigidity phenomena (such as quasi-isometric rigidity). It also allowed these techniques to be applied to the study of certain infinite groups. Research in this direction has since exploded and created the whole new field of geometric group theory.