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Regul. Chaotic Dyn., 2018, Volume 23, Issue 4, Pages 355–371 (Mi rcd328)  

This article is cited in 2 scientific papers (total in 2 papers)

Equivariant Classification of $b^m$-symplectic Surfaces

Eva Mirandaab, Arnau Planasb

a IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL University, Sorbonne Université, 77 Avenue Denfert-Rochereau, 75014 Paris, France
b 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

Abstract: Inspired by Arnold’s classification of local Poisson structures [1] in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of $b^m$-Poisson structures which can be also visualized using differential forms with singularities as $b^m$-symplectic structures. In this paper we extend the classification scheme in [24] for bm-symplectic surfaces to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct $b^m$-symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.

Keywords: Moser path method, singularities, $b^m$-symplectic manifolds, group actions

Funding Agency Grant Number
Ministerio de Economía y Competitividad de España MTM 2015-69135-P
Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Both authors are partially supported by the grants reference number MTM 2015-69135-P (MINECO/FEDER) and reference number 2017SGR932 (AGAUR).


DOI: https://doi.org/10.1134/S1560354718040019

References: PDF file   HTML file

Bibliographic databases:

MSC: 53D05, 53D17
Received: 27.10.2017
Accepted:28.05.2018
Language:

Citation: Eva Miranda, Arnau Planas, “Equivariant Classification of $b^m$-symplectic Surfaces”, Regul. Chaotic Dyn., 23:4 (2018), 355–371

Citation in format AMSBIB
\Bibitem{MirPla18}
\by Eva Miranda, Arnau Planas
\paper Equivariant Classification of $b^m$-symplectic Surfaces
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 4
\pages 355--371
\mathnet{http://mi.mathnet.ru/rcd328}
\crossref{https://doi.org/10.1134/S1560354718040019}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3836276}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000440806900001}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85051138431}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Robert Cardona, Eva Miranda, “On the Volume Elements of a Manifold with Transverse Zeroes”, Regul. Chaotic Dyn., 24:2 (2019), 187–197  mathnet  crossref
    2. R. Cardona, E. Miranda, D. Peralta-Salas, “Euler flows and singular geometric structures”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 377:2158 (2019), 20190034  crossref  mathscinet  isi  scopus
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