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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/51759
Ver los metadatos del registro completo
id |
CONICETDig_3ac13898d829b046ddd4fdf7af9a64d5 |
---|---|
oai_identifier_str |
oai:ri.conicet.gov.ar:11336/51759 |
network_acronym_str |
CONICETDig |
repository_id_str |
3498 |
network_name_str |
CONICET Digital (CONICET) |
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) |
collection |
CONICET Digital (CONICET) |
instname_str |
Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.name.fl_str_mv |
CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.mail.fl_str_mv |
dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
_version_ |
1844613312029917184 |
score |
13.070432 |