Focal radius, rigidity, and lower curvature bounds

“This is the accepted version of the following article: Luis Guijarro and Frederick Wilhelm, Focal radius, rigidity, and lower curvature bounds, which has been published in final form at: https://doi.org/10.1112/plms.12113.”

Detalles Bibliográficos
Autores: Guijarro Santamaría, Luis, Wilhelm, Frederick
Tipo de recurso: artículo
Fecha de publicación:2018
País:España
Institución:Universidad Autónoma de Madrid
Repositorio:Biblos-e Archivo. Repositorio Institucional de la UAM
Idioma:inglés
OAI Identifier:oai:repositorio.uam.es:10486/684726
Acceso en línea:http://hdl.handle.net/10486/684726
https://dx.doi.org/10.1112/plms.12113
Access Level:acceso abierto
Palabra clave:Jacobi fields
Jacobi equation
Geodesic in M
Ricci curvature
Matemáticas
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spelling Focal radius, rigidity, and lower curvature boundsGuijarro Santamaría, LuisWilhelm, FrederickJacobi fieldsJacobi equationGeodesic in MRicci curvatureMatemáticas“This is the accepted version of the following article: Luis Guijarro and Frederick Wilhelm, Focal radius, rigidity, and lower curvature bounds, which has been published in final form at: https://doi.org/10.1112/plms.12113.”We prove a new comparison lemma for Jacobi fields that exploits Wilking's transverse Jacobi equation. In contrast to standard Riccati and Jacobi comparison theorems, there are situations when our technique can be applied after the first conjugate point. Using it, we show that the focal radius of any submanifold N of positive dimension in a manifold M with sectional curvature greater than or equal to 1 does not exceed π 2 . In the case of equality, we show that N is totally geodesic in M and the universal cover of M is isometric to a sphere or a projective space with their standard metrics, provided that N is closed. Our results also hold for k th intermediate Ricci curvature, provided that the submanifold has dimension ⩾ k . Thus, in a manifold with Ricci curvature ⩾ n − 1 , all hypersurfaces have focal radius ⩽ π 2 , and space forms are the only such manifolds where equality can occur, if the submanifold is closed. Example 4.38 and Remark 5.4 show that our results cannot be proven using standard Riccati or Jacobi comparison techniquesThe first author was supported by research grants MTM2011‐22612, MTM2014‐57769‐3‐P, and MTM2017‐85934‐C3‐2‐P from the MINECO, and by ICMAT Severo Ochoa project SEV‐2015‐0554 (MINECO). This work was supported by a grant from the Simons Foundation (#358068, Frederick Wilhelm)London Mathematical SocietyDepartamento de MatemáticasFacultad de CienciasUAM. Instituto de Ciencias Matemáticas (ICMAT)20182018-02-13research articlehttp://purl.org/coar/resource_type/c_2df8fbb1VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/684726https://dx.doi.org/10.1112/plms.12113reponame:Biblos-e Archivo. Repositorio Institucional de la UAMinstname:Universidad Autónoma de MadridInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.uam.es:10486/6847262026-06-23T12:46:27Z
dc.title.none.fl_str_mv Focal radius, rigidity, and lower curvature bounds
title Focal radius, rigidity, and lower curvature bounds
spellingShingle Focal radius, rigidity, and lower curvature bounds
Guijarro Santamaría, Luis
Jacobi fields
Jacobi equation
Geodesic in M
Ricci curvature
Matemáticas
title_short Focal radius, rigidity, and lower curvature bounds
title_full Focal radius, rigidity, and lower curvature bounds
title_fullStr Focal radius, rigidity, and lower curvature bounds
title_full_unstemmed Focal radius, rigidity, and lower curvature bounds
title_sort Focal radius, rigidity, and lower curvature bounds
dc.creator.none.fl_str_mv Guijarro Santamaría, Luis
Wilhelm, Frederick
author Guijarro Santamaría, Luis
author_facet Guijarro Santamaría, Luis
Wilhelm, Frederick
author_role author
author2 Wilhelm, Frederick
author2_role author
dc.contributor.none.fl_str_mv Departamento de Matemáticas
Facultad de Ciencias
UAM. Instituto de Ciencias Matemáticas (ICMAT)
dc.subject.none.fl_str_mv Jacobi fields
Jacobi equation
Geodesic in M
Ricci curvature
Matemáticas
topic Jacobi fields
Jacobi equation
Geodesic in M
Ricci curvature
Matemáticas
description “This is the accepted version of the following article: Luis Guijarro and Frederick Wilhelm, Focal radius, rigidity, and lower curvature bounds, which has been published in final form at: https://doi.org/10.1112/plms.12113.”
publishDate 2018
dc.date.none.fl_str_mv 2018
2018-02-13
dc.type.none.fl_str_mv research article
http://purl.org/coar/resource_type/c_2df8fbb1
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10486/684726
https://dx.doi.org/10.1112/plms.12113
url http://hdl.handle.net/10486/684726
https://dx.doi.org/10.1112/plms.12113
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
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
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv London Mathematical Society
publisher.none.fl_str_mv London Mathematical Society
dc.source.none.fl_str_mv reponame:Biblos-e Archivo. Repositorio Institucional de la UAM
instname:Universidad Autónoma de Madrid
instname_str Universidad Autónoma de Madrid
reponame_str Biblos-e Archivo. Repositorio Institucional de la UAM
collection Biblos-e Archivo. Repositorio Institucional de la UAM
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300719

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