The geometry and topology of contact structures with singularities
In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-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-smoo...
| Autores: | , |
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| Tipo de recurso: | informe técnico |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/343166 |
| Acceso en línea: | https://hdl.handle.net/2117/343166 |
| Access Level: | acceso abierto |
| Palabra clave: | Contact geometry Symplectic geometry Reeb dynamics Classificació AMS::14 Algebraic geometry Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-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 $b^m$-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 those are related to smooth contact structures through a desingularization technique. The problem of existence of $b^m$-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 $b^m$-contact structure. |
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