Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics
| dc.creator | Francaviglia, S. | |
| dc.creator | Lafont, J. -F. | |
| dc.date | 2008-01-23 | |
| dc.date | 2008-01-24 | |
| dc.date.accessioned | 2026-07-07T08:56:06Z | |
| dc.date.available | 2026-07-07T08:56:06Z | |
| dc.description | 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. | |
| dc.description | 35 pages, 7 figures; modified file to orient graphics correctly | |
| dc.identifier | https://arxiv.org/abs/0801.3636 | |
| dc.identifier | http://arxiv.org/abs/0801.3636 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/146493 | |
| dc.subject | Differential Geometry | |
| dc.subject | Geometric Topology | |
| dc.subject | Metric Geometry | |
| dc.title | Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics | |
| dc.type | text |