{"id":{"repo_id":"ohiolink","oai_identifier":"oai:etd.ohiolink.edu:ucin1353088820"},"canonical_url":"https://search.dev.ndltd.org/etd/ohiolink/oai:etd.ohiolink.edu:ucin1353088820","repository":{"repo_id":"ohiolink","name":"OhioLINK","base_url":"https://etd.ohiolink.edu/acprod/odb_etd/ws/oai/oai"},"display":{"title":"Geometric Properties of the Ferrand Metric","abstract":"We investigate the differentiability of the Ferrand metric-density and find a sufficient condition that implies that the Ferrand metric-density is not differentiable and that the Ferrand Gaussian curvature is negative infinity at a point. We prove that a closed Euclidean ball, closed half-space, or the complement of an open Euclidean ball contained in a domain is convex with respect to the Ferrand metric. As a result, the arc-length parametrization of a Ferrand geodesic has Lipschitz continuous first derivative, and this result is sharp. We consider the Ferrand geometry of a convex domain. We show that Ferrand geodesics are unique, that Ferrand balls are strictly Euclidean convex, and that Ferrand geodesics can be prolonged to Ferrand geodesic rays. Finally, we study the notion of a rolling disk condition and provide a sufficient geometric condition which guarantees that the ratio of the hyperbolic and quasihyperbolic metric-densities, and also the ratio of the Ferrand and quasihyperbolic metric-densities, both approach one as an interior point approaches a boundary point.","abstract_html":"We investigate the differentiability of the Ferrand metric-density and find a sufficient condition that implies that the Ferrand metric-density is not differentiable and that the Ferrand Gaussian curvature is negative infinity at a point. We prove that a closed Euclidean ball, closed half-space, or the complement of an open Euclidean ball contained in a domain is convex with respect to the Ferrand metric. As a result, the arc-length parametrization of a Ferrand geodesic has Lipschitz continuous first derivative, and this result is sharp. We consider the Ferrand geometry of a convex domain. We show that Ferrand geodesics are unique, that Ferrand balls are strictly Euclidean convex, and that Ferrand geodesics can be prolonged to Ferrand geodesic rays. Finally, we study the notion of a rolling disk condition and provide a sufficient geometric condition which guarantees that the ratio of the hyperbolic and quasihyperbolic metric-densities, and also the ratio of the Ferrand and quasihyperbolic metric-densities, both approach one as an interior point approaches a boundary point.","abstract_has_math":false,"creators":["Julian, Poranee K."],"institution":"University of Cincinnati","degree_name":"PhD","degree_level":"doctoral","degree_discipline":"Arts and Sciences: Mathematical Sciences","degree_department":null,"school":null,"contributors":["Herron, David"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012","date_published":"2012","updated_at":"2026-07-24T03:36:23Z","subjects":["Mathematics","Ferrand","conformal metric","geodesic","Mobius invariant","geometry","hyperbolic"],"languages":["English"],"rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://rave.ohiolink.edu/etdc/view?acc_num=ucin1353088820","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Herron, David"]},{"key":"dc:creator","label":"Author","values":["Julian, Poranee K."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012"]},{"key":"dc:publisher","label":"Institution","values":["University of Cincinnati / OhioLINK"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Arts and Sciences: Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["PhD"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Cincinnati"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Ferrand","conformal metric","geodesic","Mobius invariant","geometry","hyperbolic"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. 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We show that Ferrand geodesics are unique, that Ferrand balls are strictly Euclidean convex, and that Ferrand geodesics can be prolonged to Ferrand geodesic rays. Finally, we study the notion of a rolling disk condition and provide a sufficient geometric condition which guarantees that the ratio of the hyperbolic and quasihyperbolic metric-densities, and also the ratio of the Ferrand and quasihyperbolic metric-densities, both approach one as an interior point approaches a boundary point."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf","p.130","744.7 KB"]},{"key":"dc:title","label":"Title","values":["Geometric Properties of the Ferrand Metric"]}]}],"canonical_facts":{"dc:contributor":["Herron, David"],"dc:creator":["Julian, Poranee K."],"dc:date":["2012"],"dc:description":["We investigate the differentiability of the Ferrand metric-density and find a sufficient condition that implies that the Ferrand metric-density is not differentiable and that the Ferrand Gaussian curvature is negative infinity at a point. We prove that a closed Euclidean ball, closed half-space, or the complement of an open Euclidean ball contained in a domain is convex with respect to the Ferrand metric. As a result, the arc-length parametrization of a Ferrand geodesic has Lipschitz continuous first derivative, and this result is sharp. We consider the Ferrand geometry of a convex domain. We show that Ferrand geodesics are unique, that Ferrand balls are strictly Euclidean convex, and that Ferrand geodesics can be prolonged to Ferrand geodesic rays. Finally, we study the notion of a rolling disk condition and provide a sufficient geometric condition which guarantees that the ratio of the hyperbolic and quasihyperbolic metric-densities, and also the ratio of the Ferrand and quasihyperbolic metric-densities, both approach one as an interior point approaches a boundary point."],"dc:format":["application/pdf","p.130","744.7 KB"],"dc:identifier":["http://rave.ohiolink.edu/etdc/view?acc_num=ucin1353088820"],"dc:language":["English"],"dc:publisher":["University of Cincinnati / OhioLINK"],"dc:rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"dc:subject":["Mathematics","Ferrand","conformal metric","geodesic","Mobius invariant","geometry","hyperbolic"],"dc:title":["Geometric Properties of the Ferrand Metric"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Arts and Sciences: Mathematical Sciences"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["PhD"],"thesis:institution_name":["University of Cincinnati"]},"updated_at":"2026-07-24T03:36:23Z"}