{"id":{"repo_id":"whiterose","oai_identifier":"oai:etheses.whiterose.ac.uk:1738"},"canonical_url":"https://search.dev.ndltd.org/etd/whiterose/oai:etheses.whiterose.ac.uk:1738","repository":{"repo_id":"whiterose","name":"White Rose University Consortium","base_url":"https://etheses.whiterose.ac.uk/cgi/oai2"},"display":{"title":"The Geodesic Gauss Map and Ruh-Vilms theorem for a Hypersurface in S^{n}","abstract":"We are interested to work on normal homogeneous space and in this space we calculated Live-Civita connection and we derived a useful equation 2.17. The Ruh-Vilms theorem is a statement about the Gauss map for a submanifold of R^{n+1} . Our aim is to prove, an isometrically immersed hypersurface f : M −→ S^{n} has constant mean curvature if and only if the Gauss map of γ is harmonic. Here we provide a proof of the Ruh-Vilms result using Homogeneous geometry. First shown for curves in S^{2} , then proven for a hypersurface in the n-sphere by using symmetric space identification and results in 2.17.","abstract_html":"We are interested to work on normal homogeneous space and in this space we calculated Live-Civita connection and we derived a useful equation 2.17. The Ruh-Vilms theorem is a statement about the Gauss map for a submanifold of R^{n+1} . Our aim is to prove, an isometrically immersed hypersurface f : M −→ S^{n} has constant mean curvature if and only if the Gauss map of γ is harmonic. Here we provide a proof of the Ruh-Vilms result using Homogeneous geometry. First shown for curves in S^{2} , then proven for a hypersurface in the n-sphere by using symmetric space identification and results in 2.17.","abstract_has_math":false,"creators":["Stanislaus, Mariaseelan"],"institution":"University of York","degree_name":"M.Sc by research","degree_level":"masters","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Ian , McIntosh"],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-03","date_published":"2011-03","updated_at":"2026-07-24T06:05:10Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Ian , McIntosh"]},{"key":"dc:creator","label":"Author","values":["Stanislaus, Mariaseelan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-03"]},{"key":"dc:date.issued","label":"Date","values":["2011-03"]},{"key":"dc:publisher.commercial","label":"Dc Publisher Commercial","values":["University of York"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Mathematics (York)"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of York"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://etheses.whiterose.ac.uk/id/eprint/1738/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["masters"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["M.Sc by research"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://etheses.whiterose.ac.uk/id/eprint/1738/1/MSc_thesis.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We are interested to work on normal homogeneous space and in this space we calculated Live-Civita connection and we derived a useful equation 2.17. The Ruh-Vilms theorem is a statement about the Gauss map for a submanifold of R^{n+1} . Our aim is to prove, an isometrically immersed hypersurface f : M −→ S^{n} has constant mean curvature if and only if the Gauss map of γ is harmonic. Here we provide a proof of the Ruh-Vilms result using Homogeneous geometry. First shown for curves in S^{2} , then proven for a hypersurface in the n-sphere by using symmetric space identification and results in 2.17."]},{"key":"dc:format","label":"Dc Format","values":["text"]},{"key":"dc:title","label":"Title","values":["The Geodesic Gauss Map and Ruh-Vilms theorem for a Hypersurface in S^{n}"]}]}],"canonical_facts":{"dc:contributor.advisor":["Ian , McIntosh"],"dc:creator":["Stanislaus, Mariaseelan"],"dc:date":["2011-03"],"dc:date.issued":["2011-03"],"dc:description.abstract":["We are interested to work on normal homogeneous space and in this space we calculated Live-Civita connection and we derived a useful equation 2.17. The Ruh-Vilms theorem is a statement about the Gauss map for a submanifold of R^{n+1} . Our aim is to prove, an isometrically immersed hypersurface f : M −→ S^{n} has constant mean curvature if and only if the Gauss map of γ is harmonic. Here we provide a proof of the Ruh-Vilms result using Homogeneous geometry. First shown for curves in S^{2} , then proven for a hypersurface in the n-sphere by using symmetric space identification and results in 2.17."],"dc:format":["text"],"dc:identifier.uri":["https://etheses.whiterose.ac.uk/id/eprint/1738/1/MSc_thesis.pdf"],"dc:publisher.commercial":["University of York"],"dc:publisher.department":["Mathematics (York)"],"dc:publisher.institution":["University of York"],"dc:relation.isreferencedby":["https://etheses.whiterose.ac.uk/id/eprint/1738/"],"dc:title":["The Geodesic Gauss Map and Ruh-Vilms theorem for a Hypersurface in S^{n}"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["masters"],"dc:type.qualificationname":["M.Sc by research"]},"updated_at":"2026-07-24T06:05:10Z"}