The Euclidean rank of a complete geodesic c(t) in a Riemannian manifold M is the dimension of the space of parallel Jacobi fields along c(t).  The Euclidean rank of M is the least rank of its geodesics.  Note that each geodesic c(t) has rank at least one since the velocity field c'(t) is parallel.



I'll motivate and describe the proof of the following rigidity theorem:  A complete Riemannian three-manifold has Euclidean rank at least 2 if and only if its universal covering is isometric to a Riemannian product.