The Rank Rigidity Theorem for Manifolds with No Focal Points.
dc.contributor.author | Watkins, Jordan P. | en_US |
dc.date.accessioned | 2013-09-24T16:01:46Z | |
dc.date.available | NO_RESTRICTION | en_US |
dc.date.available | 2013-09-24T16:01:46Z | |
dc.date.issued | 2013 | en_US |
dc.date.submitted | 2013 | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/99842 | |
dc.description.abstract | We say that a Riemannian manifold M has rank at least k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns-Spatzier, later generalized by Eberlein-Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank at least 2 (the "higher rank" assumption) whose isometry group G satisfies the condition that the G-recurrent vectors are dense in SM is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. We adapt the method of Ballmann and Eberlein-Heber to prove a generalization of this theorem where the manifold $M$ is assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann-Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group. | en_US |
dc.language.iso | en_US | en_US |
dc.subject | Rigidity | en_US |
dc.subject | No Focal Points | en_US |
dc.subject | Higher Rank | en_US |
dc.subject | Duality Condition | en_US |
dc.subject | Riemannian Manifolds | en_US |
dc.subject | MSC - 53C24 | en_US |
dc.title | The Rank Rigidity Theorem for Manifolds with No Focal Points. | en_US |
dc.type | Thesis | en_US |
dc.description.thesisdegreename | PhD | en_US |
dc.description.thesisdegreediscipline | Mathematics | en_US |
dc.description.thesisdegreegrantor | University of Michigan, Horace H. Rackham School of Graduate Studies | en_US |
dc.contributor.committeemember | Spatzier, Ralf J. | en_US |
dc.contributor.committeemember | Krisch, Jean P. | en_US |
dc.contributor.committeemember | Scott, G. Peter | en_US |
dc.contributor.committeemember | Canary, Richard D. | en_US |
dc.contributor.committeemember | Ji, Lizhen | en_US |
dc.subject.hlbsecondlevel | Mathematics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/99842/1/jpwatkin_1.pdf | |
dc.owningcollname | Dissertations and Theses (Ph.D. and Master's) |
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