Asymptotic Cones, Bi-lipschitz Ultraflats, and the Geometric Rank of Geodesics

Let M be a closed non-positively curved Riemannian (NPCR) manifold, M̃ its universal cover, and X an ultralimit of M̃ . For γ ⊂ M̃ a geodesic, let γω be a geodesic in X obtained as an ultralimit of γ. We show that if γω is contained in a flat in X, then the original geodesic γ supports a non-trivial, normal, parallel Jacobi field. In particular, the rank of a geodesic can be detected from the ultralimit of the universal cover. We strengthen this result by allowing for bi-Lipschitz flats satisfying certain additional hypotheses. 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.