Homogeneous hypersurfaces in symmetric spaces

A hypersurface of a Riemannian manifold is said to be (extrinsically) homogeneous if it can be obtained as an orbit of an action of a subgroup of the isometry group of the ambient space. In this case, such an action is said to be of cohomogeneity one. The study of homogeneous hypersurfaces only make...

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Detalhes bibliográficos
Autor: Otero Casal, Tomás
Tipo de documento: tese
Data de publicação:2024
País:España
Recursos:Universidad de Santiago de Compostela (USC)
Repositório:Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela
Idioma:inglês
OAI Identifier:oai:minerva.usc.gal:10347/37574
Acesso em linha:https://hdl.handle.net/10347/37574
Access Level:Acceso aberto
Palavra-chave:Homogeneous hypersurfaces
symmetric spaces
isometric actions
cohomogeneity one actions
120411 Geometría de Riemann
121008 Grupos de Lie
120109 Algebra de Lie
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spelling Homogeneous hypersurfaces in symmetric spacesOtero Casal, TomásHomogeneous hypersurfacessymmetric spacesisometric actionscohomogeneity one actions120411 Geometría de Riemann121008 Grupos de Lie120109 Algebra de LieA hypersurface of a Riemannian manifold is said to be (extrinsically) homogeneous if it can be obtained as an orbit of an action of a subgroup of the isometry group of the ambient space. In this case, such an action is said to be of cohomogeneity one. The study of homogeneous hypersurfaces only makes sense for ambient spaces with a large enough isometry group. This is the case of Riemannian symmetric spaces, which constitute an important class among Riemannian manifolds, and whose study combines ideas from various areas of mathematics like geometry, topology, algebra, and mathematical analysis. In this thesis, we tackle the classification problem for homogeneous hypersurfaces in symmetric spaces. The results can be divided into two lines. The first of these consists in the development of a structural result for cohomogeneity one actions on symmetric spaces of noncompact type. This result guarantees that any such action can be constructed by one of five standard methods, easily described in terms of Lie algebras. The second line investigates cohomogeneity one actions on products of symmetric spaces of different types. Under certain hypotheses, one can reduce the study of these actions to each factor. This allowed us to produce a classification of codimension one homogeneous foliations on simply connected symmetric spaces.Díaz Ramos, José CarlosDomínguez Vázquez, MiguelUniversidade de Santiago de Compostela. Escola de Doutoramento Internacional (EDIUS)20242024-01-0120242024-01-01doctoral thesishttp://purl.org/coar/resource_type/c_db06info:eu-repo/semantics/doctoralThesisapplication/pdfhttps://hdl.handle.net/10347/37574reponame:Minerva. Repositorio Institucional de la Universidad de Santiago de Compostelainstname:Universidad de Santiago de Compostela (USC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:minerva.usc.gal:10347/375742026-06-15T12:47:27Z
dc.title.none.fl_str_mv Homogeneous hypersurfaces in symmetric spaces
title Homogeneous hypersurfaces in symmetric spaces
spellingShingle Homogeneous hypersurfaces in symmetric spaces
Otero Casal, Tomás
Homogeneous hypersurfaces
symmetric spaces
isometric actions
cohomogeneity one actions
120411 Geometría de Riemann
121008 Grupos de Lie
120109 Algebra de Lie
title_short Homogeneous hypersurfaces in symmetric spaces
title_full Homogeneous hypersurfaces in symmetric spaces
title_fullStr Homogeneous hypersurfaces in symmetric spaces
title_full_unstemmed Homogeneous hypersurfaces in symmetric spaces
title_sort Homogeneous hypersurfaces in symmetric spaces
dc.creator.none.fl_str_mv Otero Casal, Tomás
author Otero Casal, Tomás
author_facet Otero Casal, Tomás
author_role author
dc.contributor.none.fl_str_mv Díaz Ramos, José Carlos
Domínguez Vázquez, Miguel
Universidade de Santiago de Compostela. Escola de Doutoramento Internacional (EDIUS)
dc.subject.none.fl_str_mv Homogeneous hypersurfaces
symmetric spaces
isometric actions
cohomogeneity one actions
120411 Geometría de Riemann
121008 Grupos de Lie
120109 Algebra de Lie
topic Homogeneous hypersurfaces
symmetric spaces
isometric actions
cohomogeneity one actions
120411 Geometría de Riemann
121008 Grupos de Lie
120109 Algebra de Lie
description A hypersurface of a Riemannian manifold is said to be (extrinsically) homogeneous if it can be obtained as an orbit of an action of a subgroup of the isometry group of the ambient space. In this case, such an action is said to be of cohomogeneity one. The study of homogeneous hypersurfaces only makes sense for ambient spaces with a large enough isometry group. This is the case of Riemannian symmetric spaces, which constitute an important class among Riemannian manifolds, and whose study combines ideas from various areas of mathematics like geometry, topology, algebra, and mathematical analysis. In this thesis, we tackle the classification problem for homogeneous hypersurfaces in symmetric spaces. The results can be divided into two lines. The first of these consists in the development of a structural result for cohomogeneity one actions on symmetric spaces of noncompact type. This result guarantees that any such action can be constructed by one of five standard methods, easily described in terms of Lie algebras. The second line investigates cohomogeneity one actions on products of symmetric spaces of different types. Under certain hypotheses, one can reduce the study of these actions to each factor. This allowed us to produce a classification of codimension one homogeneous foliations on simply connected symmetric spaces.
publishDate 2024
dc.date.none.fl_str_mv 2024
2024-01-01
2024
2024-01-01
dc.type.none.fl_str_mv doctoral thesis
http://purl.org/coar/resource_type/c_db06
dc.type.openaire.fl_str_mv info:eu-repo/semantics/doctoralThesis
format doctoralThesis
dc.identifier.none.fl_str_mv https://hdl.handle.net/10347/37574
url https://hdl.handle.net/10347/37574
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela
instname:Universidad de Santiago de Compostela (USC)
instname_str Universidad de Santiago de Compostela (USC)
reponame_str Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela
collection Minerva. Repositorio Institucional de la Universidad de Santiago de Compostela
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repository.mail.fl_str_mv
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