Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics

Details Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics

2008-01-23

arxiv.org/abs/0801.3636v2

Mathematics

Differential Geometry

35 pages, 7 figures; modified file to orient graphics correctly

Scientific paper

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.

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