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
Fil: Olmos, Carlos Enrique. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.
Fil: Riaño Riaño, Richar Fernando. Universidad de los Andes. Facultad de Matemáticas; Colombia.
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 ⊂ RN 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.
publishedVersion
Fil: Olmos, Carlos Enrique. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.
Fil: Riaño Riaño, Richar Fernando. Universidad de los Andes. Facultad de Matemáticas; Colombia.
Matemática Pura
Fuente
eISSN 1881-1167
Materia
Normal holonomy
Orbits of s-representations
Veronese submanifolds
Nivel de accesibilidad
acceso abierto
Condiciones de uso
Repositorio
Repositorio Digital Universitario (UNC)
Institución
Universidad Nacional de Córdoba
OAI Identificador
oai:rdu.unc.edu.ar:11086/23638

id RDUUNC_2af99916da5d0e2d3709497e62a95a7c
oai_identifier_str oai:rdu.unc.edu.ar:11086/23638
network_acronym_str RDUUNC
repository_id_str 2572
network_name_str Repositorio Digital Universitario (UNC)
spelling Normal holonomy of orbits and Veronese submanifoldsOlmos, Carlos EnriqueRiaño Riaño, Richar FernandoNormal holonomyOrbits of s-representationsVeronese submanifoldsFil: Olmos, Carlos Enrique. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Riaño Riaño, Richar Fernando. Universidad de los Andes. Facultad de Matemáticas; Colombia.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 ⊂ RN 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.publishedVersionFil: Olmos, Carlos Enrique. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Riaño Riaño, Richar Fernando. Universidad de los Andes. Facultad de Matemáticas; Colombia.Matemática Pura2015info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfOlmos, C. E. y Riaño Riaño, R. F. (2015). Normal holonomy of orbits and Veronese submanifolds. Journal of the Mathematical Society of Japan, 67 (3), 903-942. https://doi.org/10.2969/jmsj/06730903http://hdl.handle.net/11086/23638https://doi.org/10.2969/jmsj/06730903eISSN 1881-1167reponame:Repositorio Digital Universitario (UNC)instname:Universidad Nacional de Córdobainstacron:UNCenginfo:eu-repo/semantics/openAccess2025-09-29T13:44:20Zoai:rdu.unc.edu.ar:11086/23638Institucionalhttps://rdu.unc.edu.ar/Universidad públicaNo correspondehttp://rdu.unc.edu.ar/oai/snrdoca.unc@gmail.comArgentinaNo correspondeNo correspondeNo correspondeopendoar:25722025-09-29 13:44:21.065Repositorio Digital Universitario (UNC) - Universidad Nacional de Córdobafalse
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
dc.description.none.fl_txt_mv Fil: Olmos, Carlos Enrique. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.
Fil: Riaño Riaño, Richar Fernando. Universidad de los Andes. Facultad de Matemáticas; Colombia.
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 ⊂ RN 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.
publishedVersion
Fil: Olmos, Carlos Enrique. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.
Fil: Riaño Riaño, Richar Fernando. Universidad de los Andes. Facultad de Matemáticas; Colombia.
Matemática Pura
description Fil: Olmos, Carlos Enrique. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.
publishDate 2015
dc.date.none.fl_str_mv 2015
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 Olmos, C. E. y Riaño Riaño, R. F. (2015). Normal holonomy of orbits and Veronese submanifolds. Journal of the Mathematical Society of Japan, 67 (3), 903-942. https://doi.org/10.2969/jmsj/06730903
http://hdl.handle.net/11086/23638
https://doi.org/10.2969/jmsj/06730903
identifier_str_mv Olmos, C. E. y Riaño Riaño, R. F. (2015). Normal holonomy of orbits and Veronese submanifolds. Journal of the Mathematical Society of Japan, 67 (3), 903-942. https://doi.org/10.2969/jmsj/06730903
url http://hdl.handle.net/11086/23638
https://doi.org/10.2969/jmsj/06730903
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv eISSN 1881-1167
reponame:Repositorio Digital Universitario (UNC)
instname:Universidad Nacional de Córdoba
instacron:UNC
reponame_str Repositorio Digital Universitario (UNC)
collection Repositorio Digital Universitario (UNC)
instname_str Universidad Nacional de Córdoba
instacron_str UNC
institution UNC
repository.name.fl_str_mv Repositorio Digital Universitario (UNC) - Universidad Nacional de Córdoba
repository.mail.fl_str_mv oca.unc@gmail.com
_version_ 1844618980104339456
score 13.070432