Regular and Chaotic Dynamics

, Volume 24, Issue 2, pp 187–197 | Cite as

On the Volume Elements of a Manifold with Transverse Zeroes

  • Robert CardonaEmail author
  • Eva Miranda


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.


Moser path method volume forms singularities b-symplectic manifolds 

MSC2010 numbers

53D05 53D17 


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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.BGSMath Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Edifici P, UPCUniversitat Politècnica de Catalunya and Barcelona Graduate School of MathematicsBarcelonaSpain
  2. 2.Observatoire de Paris Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Edifici P, UPCUniversitat Politècnica de Catalunya, Barcelona Graduate School of Mathematics BGSMath, Instituto de Ciencias Matemáticas ICMATBarcelonaSpain

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