Fixing K=-1, 0, or 1, a Riemannian manifold (M, g) is said to have higher hyperbolic, spherical, or Euclidean rank if every geodesic in M admits a normal parallel field making curvature K with the geodesic. Rank rigidity results, usually involves a priori sectional curvature bounds, characterize locally symmetric spaces in terms of these geometric notions of rank.

After giving a short survey of historical results, I’ll discuss how rank rigidity holds in dimension three without a priori sectional curvature bounds.