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This article is cited in 5 scientific papers (total in 5 papers)
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
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 $b^m$-Nambu structures.
Keywords:
Moser path method, volume forms, singularities, $b$-symplectic manifolds.
Received: 10.12.2018 Accepted: 25.02.2019
Citation:
Robert Cardona, Eva Miranda, “On the Volume Elements of a Manifold with Transverse Zeroes”, Regul. Chaotic Dyn., 24:2 (2019), 187–197
Linking options:
https://www.mathnet.ru/eng/rcd452 https://www.mathnet.ru/eng/rcd/v24/i2/p187
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Abstract page: | 219 | References: | 34 |
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