Persistent Link:
http://hdl.handle.net/10150/193749
Title:
Geodesics of Random Riemannian Metrics
Author:
LaGatta, Tom
Issue Date:
2010
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on R^d . We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction v, the geodesic starting from the origin in the direction v is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP.
Type:
text; Electronic Dissertation
Keywords:
first-passage percolation; motion in a random environment; riemannian geometry
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Wehr, Jan
Committee Chair:
Wehr, Jan

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleGeodesics of Random Riemannian Metricsen_US
dc.creatorLaGatta, Tomen_US
dc.contributor.authorLaGatta, Tomen_US
dc.date.issued2010en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractWe introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on R^d . We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction v, the geodesic starting from the origin in the direction v is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectfirst-passage percolationen_US
dc.subjectmotion in a random environmenten_US
dc.subjectriemannian geometryen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorWehr, Janen_US
dc.contributor.chairWehr, Janen_US
dc.contributor.committeememberWehr, Janen_US
dc.contributor.committeememberGlickenstein, Daviden_US
dc.contributor.committeememberKennedy, Thomas G.en_US
dc.contributor.committeememberWatkins, Joseph C.en_US
dc.identifier.proquest11083en_US
dc.identifier.oclc659755021en_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.