{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20441"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20441","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Geometric and dynamical properties of Riemannian foliations","abstract":"Given a Riemannian foliation ${\\cal F}$ on a Riemannian manifold M with a bundle-like metric, geometric and dynamical properties of geodesics orthogonal to the leaves of the foliation are studied.","abstract_html":"Given a Riemannian foliation ${\\cal F}$ on a Riemannian manifold M with a bundle-like metric, geometric and dynamical properties of geodesics orthogonal to the leaves of the foliation are studied.","abstract_has_math":true,"creators":["Kim, Hobum"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Bishop, Richard L."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:39:17Z","date_published":"2011-05-07T12:39:17Z","updated_at":"2026-07-22T22:25:16Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Kim, Hobum"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114294","(UMI)AAI9114294"],"render_values":[{"text":"AAI9114294","href":null,"code":true},{"text":"(UMI)AAI9114294","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20441","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bishop, Richard L."]},{"key":"dc:creator","label":"Author","values":["Kim, Hobum"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:39:17Z","10000-01-01","1990"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 Kim, Hobum"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114294","(UMI)AAI9114294","http://hdl.handle.net/2142/20441"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Given a Riemannian foliation ${\\cal F}$ on a Riemannian manifold M with a bundle-like metric, geometric and dynamical properties of geodesics orthogonal to the leaves of the foliation are studied.","In one line of work, the concepts of ${\\cal F}$-Jacobi fields and ${\\cal F}$-Jacobi tensors are introduced. Using these concepts, an upper bound for the index of a focal point of a leaf is obtained, when the orthogonal complement of the foliation is involutive. In particular, it is proved that there is no focal point of a leaf of a Riemannian foliation of codimension one. Moreover, it is also proved that if M is a complete Riemannian manifold of nonnegative sectional curvature, and if the norm of the integrability tensor is small compared with the sectional curvature of M, then ${\\cal F}$ is totally geodesic.","In another line of work, it is proved that: (1) ${\\cal F}$ is harmonic if and only if the geodesic flow preserves the corresponding Riemannian volume form on the normal bundle corresponding to a Sasaki-type metric, in case ${\\cal F}$ is transversally flat.","(2) There is no Riemannian foliation on a compact Riemannian manifold of negative sectional curvature. The proof uses Oseledec's multiplicative ergodic theorem.","Made available in DSpace on 2011-05-07T12:39:17Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114294.pdf: 1735927 bytes, checksum: afc4609014734ac2b7aa4ba1b2c2e04b (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:43:55Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:19:16-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Geometric and dynamical properties of Riemannian foliations"]}]}],"canonical_facts":{"dc:contributor":["Bishop, Richard L."],"dc:creator":["Kim, Hobum"],"dc:date":["2011-05-07T12:39:17Z","10000-01-01","1990"],"dc:description":["Given a Riemannian foliation ${\\cal F}$ on a Riemannian manifold M with a bundle-like metric, geometric and dynamical properties of geodesics orthogonal to the leaves of the foliation are studied.","In one line of work, the concepts of ${\\cal F}$-Jacobi fields and ${\\cal F}$-Jacobi tensors are introduced. Using these concepts, an upper bound for the index of a focal point of a leaf is obtained, when the orthogonal complement of the foliation is involutive. In particular, it is proved that there is no focal point of a leaf of a Riemannian foliation of codimension one. Moreover, it is also proved that if M is a complete Riemannian manifold of nonnegative sectional curvature, and if the norm of the integrability tensor is small compared with the sectional curvature of M, then ${\\cal F}$ is totally geodesic.","In another line of work, it is proved that: (1) ${\\cal F}$ is harmonic if and only if the geodesic flow preserves the corresponding Riemannian volume form on the normal bundle corresponding to a Sasaki-type metric, in case ${\\cal F}$ is transversally flat.","(2) There is no Riemannian foliation on a compact Riemannian manifold of negative sectional curvature. The proof uses Oseledec's multiplicative ergodic theorem.","Made available in DSpace on 2011-05-07T12:39:17Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114294.pdf: 1735927 bytes, checksum: afc4609014734ac2b7aa4ba1b2c2e04b (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:43:55Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:19:16-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9114294","(UMI)AAI9114294","http://hdl.handle.net/2142/20441"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Kim, Hobum"],"dc:subject":["Mathematics"],"dc:title":["Geometric and dynamical properties of Riemannian foliations"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:16Z"}