Normal holonomy of orbits and Veronese submanifolds

Autores
Olmos, Carlos Enrique; Riaño Riaño, Richar Fernando
Año de publicación
2015
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let Mn, n≥2, be a full and irreducible homogeneous submanifold of the sphere SN-1⊂ ℝ N such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then Mn must be an orbit of an irreducible s-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space). If n = 2, then the normal holonomy is always transitive, unless M is a homogeneous isoparametric hypersurface of the sphere (and so the conjecture is true in this case). We prove the conjecture when n = 3. In this case M3 must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that use information from two different fibrations with the same total space (the holonomy tube and the caustic fibrations). We also prove the conjecture for n ≥3 when the normal holonomy acts irreducibly and the codimension is the maximal possible n(n+1)=2. This gives a characterization of Veronese submanifolds in terms of normal holonomy. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality (in the sphere). Another result of the paper, used for the case n = 3, is that the number of irreducible factors of the local normal holonomy group, for any Euclidean submanifold Mn, is less or equal than [n=2] (which is the rank of the orthogonal group SO(n)). This bound is sharp and improves the known bound n(n-1)/2.
Fil: Olmos, Carlos Enrique. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Fil: Riaño Riaño, Richar Fernando. Universidad de los Andes; Colombia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Materia
Normal Holonomy
Orbits of S-Representations
Veronese Submanifolds
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/51759

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spelling Normal holonomy of orbits and Veronese submanifoldsOlmos, Carlos EnriqueRiaño Riaño, Richar FernandoNormal HolonomyOrbits of S-RepresentationsVeronese Submanifoldshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let Mn, n≥2, be a full and irreducible homogeneous submanifold of the sphere SN-1⊂ ℝ N such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then Mn must be an orbit of an irreducible s-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space). If n = 2, then the normal holonomy is always transitive, unless M is a homogeneous isoparametric hypersurface of the sphere (and so the conjecture is true in this case). We prove the conjecture when n = 3. In this case M3 must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that use information from two different fibrations with the same total space (the holonomy tube and the caustic fibrations). We also prove the conjecture for n ≥3 when the normal holonomy acts irreducibly and the codimension is the maximal possible n(n+1)=2. This gives a characterization of Veronese submanifolds in terms of normal holonomy. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality (in the sphere). Another result of the paper, used for the case n = 3, is that the number of irreducible factors of the local normal holonomy group, for any Euclidean submanifold Mn, is less or equal than [n=2] (which is the rank of the orthogonal group SO(n)). This bound is sharp and improves the known bound n(n-1)/2.Fil: Olmos, Carlos Enrique. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Riaño Riaño, Richar Fernando. Universidad de los Andes; Colombia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaMath Soc Japan2015-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/51759Olmos, Carlos Enrique; Riaño Riaño, Richar Fernando; Normal holonomy of orbits and Veronese submanifolds; Math Soc Japan; Journal Of The Mathematical Society Of Japan; 67; 3; 6-2015; 903-9420025-5645CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://projecteuclid.org/euclid.jmsj/1438777435info:eu-repo/semantics/altIdentifier/doi/10.2969/jmsj/06730903info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:41:32Zoai:ri.conicet.gov.ar:11336/51759instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 09:41:33.484CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Normal holonomy of orbits and Veronese submanifolds
title Normal holonomy of orbits and Veronese submanifolds
spellingShingle Normal holonomy of orbits and Veronese submanifolds
Olmos, Carlos Enrique
Normal Holonomy
Orbits of S-Representations
Veronese Submanifolds
title_short Normal holonomy of orbits and Veronese submanifolds
title_full Normal holonomy of orbits and Veronese submanifolds
title_fullStr Normal holonomy of orbits and Veronese submanifolds
title_full_unstemmed Normal holonomy of orbits and Veronese submanifolds
title_sort Normal holonomy of orbits and Veronese submanifolds
dc.creator.none.fl_str_mv Olmos, Carlos Enrique
Riaño Riaño, Richar Fernando
author Olmos, Carlos Enrique
author_facet Olmos, Carlos Enrique
Riaño Riaño, Richar Fernando
author_role author
author2 Riaño Riaño, Richar Fernando
author2_role author
dc.subject.none.fl_str_mv Normal Holonomy
Orbits of S-Representations
Veronese Submanifolds
topic Normal Holonomy
Orbits of S-Representations
Veronese Submanifolds
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let Mn, n≥2, be a full and irreducible homogeneous submanifold of the sphere SN-1⊂ ℝ N such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then Mn must be an orbit of an irreducible s-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space). If n = 2, then the normal holonomy is always transitive, unless M is a homogeneous isoparametric hypersurface of the sphere (and so the conjecture is true in this case). We prove the conjecture when n = 3. In this case M3 must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that use information from two different fibrations with the same total space (the holonomy tube and the caustic fibrations). We also prove the conjecture for n ≥3 when the normal holonomy acts irreducibly and the codimension is the maximal possible n(n+1)=2. This gives a characterization of Veronese submanifolds in terms of normal holonomy. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality (in the sphere). Another result of the paper, used for the case n = 3, is that the number of irreducible factors of the local normal holonomy group, for any Euclidean submanifold Mn, is less or equal than [n=2] (which is the rank of the orthogonal group SO(n)). This bound is sharp and improves the known bound n(n-1)/2.
Fil: Olmos, Carlos Enrique. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Fil: Riaño Riaño, Richar Fernando. Universidad de los Andes; Colombia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
description It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let Mn, n≥2, be a full and irreducible homogeneous submanifold of the sphere SN-1⊂ ℝ N such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then Mn must be an orbit of an irreducible s-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space). If n = 2, then the normal holonomy is always transitive, unless M is a homogeneous isoparametric hypersurface of the sphere (and so the conjecture is true in this case). We prove the conjecture when n = 3. In this case M3 must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that use information from two different fibrations with the same total space (the holonomy tube and the caustic fibrations). We also prove the conjecture for n ≥3 when the normal holonomy acts irreducibly and the codimension is the maximal possible n(n+1)=2. This gives a characterization of Veronese submanifolds in terms of normal holonomy. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality (in the sphere). Another result of the paper, used for the case n = 3, is that the number of irreducible factors of the local normal holonomy group, for any Euclidean submanifold Mn, is less or equal than [n=2] (which is the rank of the orthogonal group SO(n)). This bound is sharp and improves the known bound n(n-1)/2.
publishDate 2015
dc.date.none.fl_str_mv 2015-06
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/51759
Olmos, Carlos Enrique; Riaño Riaño, Richar Fernando; Normal holonomy of orbits and Veronese submanifolds; Math Soc Japan; Journal Of The Mathematical Society Of Japan; 67; 3; 6-2015; 903-942
0025-5645
CONICET Digital
CONICET
url http://hdl.handle.net/11336/51759
identifier_str_mv Olmos, Carlos Enrique; Riaño Riaño, Richar Fernando; Normal holonomy of orbits and Veronese submanifolds; Math Soc Japan; Journal Of The Mathematical Society Of Japan; 67; 3; 6-2015; 903-942
0025-5645
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://projecteuclid.org/euclid.jmsj/1438777435
info:eu-repo/semantics/altIdentifier/doi/10.2969/jmsj/06730903
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Math Soc Japan
publisher.none.fl_str_mv Math Soc Japan
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
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