Abstract:
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 bm-Nambu structures.
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.
\Bibitem{CarMir19}
\by Robert Cardona, Eva Miranda
\paper On the Volume Elements of a Manifold with Transverse Zeroes
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 2
\pages 187--197
\mathnet{http://mi.mathnet.ru/rcd452}
\crossref{https://doi.org/10.1134/S1560354719020047}
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A. Yu. Konyaev, E. A. Kudryavtseva, V. I. Sidel'nikov, “Geometry and topology of two-dimensional symplectic manifolds with generic singularities and Hamiltonian systems on them”, Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 79:5 (2024), 230–243
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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”, Rus. J. Nonlin. Dyn., 16:2 (2020), 325–341
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
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