On the Volume Elements of a Manifold with Transverse Zeroes

    2019, Volume 24, Number 2, pp.  187-197

    Author(s): Cardona R., Miranda E.

    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
    Citation: Cardona R., Miranda E., On the Volume Elements of a Manifold with Transverse Zeroes, Regular and Chaotic Dynamics, 2019, Volume 24, Number 2, pp. 187-197

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