  Given a geodesic inside a simply-connected, complete, non-positively curved
Riemannian (NPCR) manifold M, we get an associated geodesic inside the
asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is
contained inside a bi-Lipschitz flat, then the original geodesic supports a
non-trivial, orthogonal, parallel Jacobi field. As applications we obtain (1)
constraints on the behavior of quasi-isometries between complete, simply
connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by
certain manifolds, and (3) a correspondence between metric splittings of
complete, simply connected NPCR manifolds, and metric splittings of its
asymptotic cones. Furthermore, combining our results with the
Ballmann-Burns-Spatzier rigidity theorem and the classic Mostow rigidity, we
also obtain (4) a new proof of Gromov's rigidity theorem for higher rank
locally symmetric spaces.
