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

Autor:	Francaviglia, S., Lafont, J. -F.
Rok vydání:	2008
Předmět:	Mathematics - Differential Geometry Mathematics - Geometric Topology Mathematics - Metric Geometry
Druh dokumentu:	Working Paper
Popis:	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. Comment: 35 pages, 7 figures; modified file to orient graphics correctly
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/0801.3636 Zobrazit plný text záznamu View this record from Arxiv