On the volume elements of a manifold with transverse zeroes
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Moser proved in 1965 in his seminal paper  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  and extend it to bm-Nambu structures.
CitationMiranda, E.; Cardona, R. On the volume elements of a manifold with transverse zeroes. "Regular and chaotic dynamics", 2019, vol. 24, núm. Issue 2, p. 187-197.