{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71220"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71220","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Immersions of Positively Curved Manifolds Into Manifolds With Curvature Bounded Above","abstract":"This paper investigates the following questions. If a compact manifold, M, with positive sectional curvature is isometrically immersed in some ambient space, N, what is the radius of the smallest ball in which its image lies? Additionally, when can the knowledge that the image lies inside a ball of restricted size be used to conclude that the immersion is an imbedding whose image bounds a convex body in N?","abstract_html":"This paper investigates the following questions. If a compact manifold, M, with positive sectional curvature is isometrically immersed in some ambient space, N, what is the radius of the smallest ball in which its image lies? Additionally, when can the knowledge that the image lies inside a ball of restricted size be used to conclude that the immersion is an imbedding whose image bounds a convex body in N?","abstract_has_math":false,"creators":["Menninga, Nadine Louise"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:10Z","date_published":"2014-12-16T06:18:10Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8422134"],"render_values":[{"text":"(UMI)AAI8422134","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71220","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Menninga, Nadine Louise"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:10Z","10000-01-01","1984"]},{"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":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71220","(UMI)AAI8422134"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This paper investigates the following questions. If a compact manifold, M, with positive sectional curvature is isometrically immersed in some ambient space, N, what is the radius of the smallest ball in which its image lies? Additionally, when can the knowledge that the image lies inside a ball of restricted size be used to conclude that the immersion is an imbedding whose image bounds a convex body in N?","The major results presented in this paper are the following theorems:","Theorem 1. Let M be a compact, connected Riemannian manifold of dimension m, m (GREATERTHEQ) 2, with K(M) (GREATERTHEQ) 1/c('2), where c is a positive constant. Let N be an n-dimensional Riemannian manifold such that (pi)c (LESSTHEQ) i(N) and K(N) ) N is an isometric immersion, then x(M) is contained in a metric ball of N with radius R 0 and simply connected space N of constant curvature less than 1/(4c('2)), there exists an M satisfying the above conditions and an immersion x: M (--->) M such that x(M) lies in no ball of radius 1/2(pi)c - (epsilon).","Theorem 2. Let M be a compact, connected, orientable, Riemannian manifold of dimension n-1, with n-1 (GREATERTHEQ) 2, and K(M) (GREATERTHEQ) 1/c('2), c is a positive constant. Let N be an n-dimensional Riemannian manifold such that (pi)c (LESSTHEQ) i(N) and K(N) ) N be an isometric immersion. Then x imbeds M into N as the boundary of a convex body.","In the above theorems K(N) is the sectional curvature function of the manifold N and i(N) is the injectivity radius of the manifold N.","Made available in DSpace on 2014-12-16T06:18:10Z (GMT). No. of bitstreams: 1 8422134.pdf: 1124133 bytes, checksum: 4abb6cea43187b9cf419100ff763e09d (MD5) Previous issue date: 1984","Embargo set by: Seth Robbins for item 71386 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","40 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984."]},{"key":"dc:title","label":"Title","values":["Immersions of Positively Curved Manifolds Into Manifolds With Curvature Bounded Above"]}]}],"canonical_facts":{"dc:creator":["Menninga, Nadine Louise"],"dc:date":["2014-12-16T06:18:10Z","10000-01-01","1984"],"dc:description":["This paper investigates the following questions. If a compact manifold, M, with positive sectional curvature is isometrically immersed in some ambient space, N, what is the radius of the smallest ball in which its image lies? Additionally, when can the knowledge that the image lies inside a ball of restricted size be used to conclude that the immersion is an imbedding whose image bounds a convex body in N?","The major results presented in this paper are the following theorems:","Theorem 1. Let M be a compact, connected Riemannian manifold of dimension m, m (GREATERTHEQ) 2, with K(M) (GREATERTHEQ) 1/c('2), where c is a positive constant. Let N be an n-dimensional Riemannian manifold such that (pi)c (LESSTHEQ) i(N) and K(N) ) N is an isometric immersion, then x(M) is contained in a metric ball of N with radius R 0 and simply connected space N of constant curvature less than 1/(4c('2)), there exists an M satisfying the above conditions and an immersion x: M (--->) M such that x(M) lies in no ball of radius 1/2(pi)c - (epsilon).","Theorem 2. Let M be a compact, connected, orientable, Riemannian manifold of dimension n-1, with n-1 (GREATERTHEQ) 2, and K(M) (GREATERTHEQ) 1/c('2), c is a positive constant. Let N be an n-dimensional Riemannian manifold such that (pi)c (LESSTHEQ) i(N) and K(N) ) N be an isometric immersion. Then x imbeds M into N as the boundary of a convex body.","In the above theorems K(N) is the sectional curvature function of the manifold N and i(N) is the injectivity radius of the manifold N.","Made available in DSpace on 2014-12-16T06:18:10Z (GMT). No. of bitstreams: 1 8422134.pdf: 1124133 bytes, checksum: 4abb6cea43187b9cf419100ff763e09d (MD5) Previous issue date: 1984","Embargo set by: Seth Robbins for item 71386 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","40 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984."],"dc:identifier":["http://hdl.handle.net/2142/71220","(UMI)AAI8422134"],"dc:subject":["Mathematics"],"dc:title":["Immersions of Positively Curved Manifolds Into Manifolds With Curvature Bounded Above"],"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":"2026-07-22T22:26:04Z"}