The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday

Autores
Olmos, Carlos; Reggiani, Silvio Nicolás
Año de publicación
2012
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul- Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split o? locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang-Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (except for spheres, or for compact simple Lie groups with a bi-invariant metric).
Fil: Olmos, Carlos. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
Fil: Reggiani, Silvio Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; Argentina
Materia
HOLONOMY
SKEW-TORSION
NATURALLY REDUCTIVE
ISOMETRY GROUP
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:11336/195607

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spelling The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthdayOlmos, CarlosReggiani, Silvio NicolásHOLONOMYSKEW-TORSIONNATURALLY REDUCTIVEISOMETRY GROUPhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul- Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split o? locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang-Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (except for spheres, or for compact simple Lie groups with a bi-invariant metric).Fil: Olmos, Carlos. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaFil: Reggiani, Silvio Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaDe Gruyter2012-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/195607Olmos, Carlos; Reggiani, Silvio Nicolás; The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday; De Gruyter; Journal Fur Die Reine Und Angewandte Mathematik; 664; 3-2012; 29-530075-4102CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1515/CRELLE.2011.100info:eu-repo/semantics/altIdentifier/url/https://www.degruyter.com/document/doi/10.1515/CRELLE.2011.100/htmlinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:46:09Zoai:ri.conicet.gov.ar:11336/195607instacron: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 10:46:09.476CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday
title The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday
spellingShingle The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday
Olmos, Carlos
HOLONOMY
SKEW-TORSION
NATURALLY REDUCTIVE
ISOMETRY GROUP
title_short The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday
title_full The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday
title_fullStr The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday
title_full_unstemmed The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday
title_sort The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday
dc.creator.none.fl_str_mv Olmos, Carlos
Reggiani, Silvio Nicolás
author Olmos, Carlos
author_facet Olmos, Carlos
Reggiani, Silvio Nicolás
author_role author
author2 Reggiani, Silvio Nicolás
author2_role author
dc.subject.none.fl_str_mv HOLONOMY
SKEW-TORSION
NATURALLY REDUCTIVE
ISOMETRY GROUP
topic HOLONOMY
SKEW-TORSION
NATURALLY REDUCTIVE
ISOMETRY GROUP
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul- Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split o? locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang-Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (except for spheres, or for compact simple Lie groups with a bi-invariant metric).
Fil: Olmos, Carlos. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
Fil: Reggiani, Silvio Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; Argentina
description We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul- Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split o? locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang-Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (except for spheres, or for compact simple Lie groups with a bi-invariant metric).
publishDate 2012
dc.date.none.fl_str_mv 2012-03
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/11336/195607
Olmos, Carlos; Reggiani, Silvio Nicolás; The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday; De Gruyter; Journal Fur Die Reine Und Angewandte Mathematik; 664; 3-2012; 29-53
0075-4102
CONICET Digital
CONICET
url http://hdl.handle.net/11336/195607
identifier_str_mv Olmos, Carlos; Reggiani, Silvio Nicolás; The skew-torsion holonomy theorem and naturally reductive spaces: To Isabel Dotti and Roberto Miatello on the occasion of their birthday; De Gruyter; Journal Fur Die Reine Und Angewandte Mathematik; 664; 3-2012; 29-53
0075-4102
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1515/CRELLE.2011.100
info:eu-repo/semantics/altIdentifier/url/https://www.degruyter.com/document/doi/10.1515/CRELLE.2011.100/html
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 De Gruyter
publisher.none.fl_str_mv De Gruyter
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
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