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Regul. Chaotic Dyn., 2019, том 24, выпуск 2, страницы 187–197 (Mi rcd452)  

Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)

On the Volume Elements of a Manifold with Transverse Zeroes

Robert Cardonaa, Eva Mirandab

a Universitat Politècnica de Catalunya and Barcelona Graduate School of Mathematics, BGSMath, Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Edifici P, UPC, Avinguda del Doctor Marañon 44-50 08028, Barcelona, Spain
b Universitat Politècnica de Catalunya, Barcelona Graduate School of Mathematics BGSMath, Instituto de Ciencias Matemáticas ICMAT, Observatoire de Paris, Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Edifici P, UPC, Avinguda del Doctor Marañon 44-50 08028, Barcelona, Spain

Аннотация: Moser proved in 1965 in his seminal paper [15] that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from a hypersurface where they fulfill a transversality assumption ($b$-Poisson structures). We do this using the desingularization technique introduced in [10] and extend it to $b^m$-Nambu structures.

Ключевые слова: Moser path method, volume forms, singularities, $b$-symplectic manifolds

Финансовая поддержка Номер гранта
Ministry of Science and Innovation of Spanish MDM-2014-0445
Ministerio de Economía y Competitividad de España MTM2015-69135-P
Agència de Gestiö d'Ajuts Universitaris i de Recerca 2017SGR932
National Science Foundation DMS-1440140
E. Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). Both authors are supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR). Part of the work that led to this paper took place at the Fields Institute in Toronto, while the second author was Invited Professor during the Focus Program on Poisson Geometry and Physics in July 2018. 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.


DOI: https://doi.org/10.1134/S1560354719020047

Список литературы: PDF файл   HTML файл

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Тип публикации: Статья
MSC: 53D05, 53D17
Поступила в редакцию: 10.12.2018
Принята в печать:25.02.2019
Язык публикации: английский

Образец цитирования: Robert Cardona, Eva Miranda, “On the Volume Elements of a Manifold with Transverse Zeroes”, Regul. Chaotic Dyn., 24:2 (2019), 187–197

Цитирование в формате AMSBIB
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\by Robert Cardona, Eva Miranda
\paper On the Volume Elements of a Manifold with Transverse Zeroes
\jour Regul. Chaotic Dyn.
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\vol 24
\issue 2
\pages 187--197
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\crossref{https://doi.org/10.1134/S1560354719020047}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    Эта публикация цитируется в следующих статьяx:
    1. Cardona R., Miranda E., Peralta-Salas D., “Euler Flows and Singular Geometric Structures”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 377:2158 (2019), 20190034  crossref  mathscinet  isi  scopus
    2. V. V. Kuzenov, S. V. Ryzhkov, A. V. Starostin, “Development of a Mathematical Model and the Numerical Solution Method in a Combined Impact Scheme for MIF Target”, Нелинейная динам., 16:2 (2020), 325–341  mathnet  crossref
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