Contact structures with singularities: From local to global

In this article we introduce and analyze in detail singular contact structures, with an emphasis on bm-contact structures, which are tangent to a given smooth hypersurface Z and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth di...

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Detalles Bibliográficos
Autores: Miranda, E., Oms, C.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2023
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2072/536853
Acceso en línea:http://hdl.handle.net/2072/536853
Access Level:acceso abierto
Palabra clave:b-Symplectic manifolds
Contact structures
Jacobi manifolds
Singularities
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spelling Contact structures with singularities: From local to globalMiranda, E.Oms, C.b-Symplectic manifoldsContact structuresJacobi manifoldsSingularitiesIn this article we introduce and analyze in detail singular contact structures, with an emphasis on bm-contact structures, which are tangent to a given smooth hypersurface Z and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called bm-contact forms, having an associated critical hypersurface Z. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of manifolds endowed with such singular contact forms are related to smooth contact structures via desingularization. The problem of existence of bm-contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a bm-contact structure. In particular, given an almost contact manifold M with a hypersurface Z, this yields the existence of a b2k-contact structure on M realizing Z as a critical set. As a consequence of the desingularization techniques in [21], we prove the existence of folded contact forms on any almost contact manifold. © 2023 The Author(s)Eva Miranda and Cédric Oms are partially supported by the AEI grant PID2019-103849GB-I00 of MCIN/AEI/10.13039/501100011033. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021, by AGAUR via the 2021 SGR 00603 grant Geometria de Varietats i Aplicacions, GEOMVAP Geometry of Manifolds and Applications, GEOMVAP and by the Spanish State Research Agency , through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M ). Eva Miranda was supported by a Chaire d'Excellence of the Fondation Sciences Mathématiques de Paris when this project started and this work has been supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d'Avenir” program (reference: ANR-10-LABX-0098 ). This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester. Cédric Oms is partially supported by the project ANR CoSyDy ( ANR-CE40-0014 ).Elsevier B.V.2023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion22 p.application/pdfhttp://hdl.handle.net/2072/536853RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésL'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: http://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/5368532026-05-29T05:05:01Z
dc.title.none.fl_str_mv Contact structures with singularities: From local to global
title Contact structures with singularities: From local to global
spellingShingle Contact structures with singularities: From local to global
Miranda, E.
b-Symplectic manifolds
Contact structures
Jacobi manifolds
Singularities
title_short Contact structures with singularities: From local to global
title_full Contact structures with singularities: From local to global
title_fullStr Contact structures with singularities: From local to global
title_full_unstemmed Contact structures with singularities: From local to global
title_sort Contact structures with singularities: From local to global
dc.creator.none.fl_str_mv Miranda, E.
Oms, C.
author Miranda, E.
author_facet Miranda, E.
Oms, C.
author_role author
author2 Oms, C.
author2_role author
dc.subject.none.fl_str_mv b-Symplectic manifolds
Contact structures
Jacobi manifolds
Singularities
topic b-Symplectic manifolds
Contact structures
Jacobi manifolds
Singularities
description In this article we introduce and analyze in detail singular contact structures, with an emphasis on bm-contact structures, which are tangent to a given smooth hypersurface Z and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called bm-contact forms, having an associated critical hypersurface Z. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of manifolds endowed with such singular contact forms are related to smooth contact structures via desingularization. The problem of existence of bm-contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a bm-contact structure. In particular, given an almost contact manifold M with a hypersurface Z, this yields the existence of a b2k-contact structure on M realizing Z as a critical set. As a consequence of the desingularization techniques in [21], we prove the existence of folded contact forms on any almost contact manifold. © 2023 The Author(s)
publishDate 2023
dc.date.none.fl_str_mv 2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/2072/536853
url http://hdl.handle.net/2072/536853
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 22 p.
application/pdf
dc.publisher.none.fl_str_mv Elsevier B.V.
publisher.none.fl_str_mv Elsevier B.V.
dc.source.none.fl_str_mv RECERCAT (Dipòsit de la Recerca de Catalunya)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
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