Flat Bi-Hamiltonian Structures and Invariant Densities
نویسنده
چکیده
A bi-Hamiltonian structure is a pair of Poisson structures P , Q which are compatible, meaning that any linear combination αP+βQ is again a Poisson structure. A biHamiltonian structure (P,Q) is called flat if P and Q can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic biHamiltonian structure (P,Q) on an odd-dimensional manifold is flat if and only if there exists a local density which is preserved by all vector fields Hamiltonian with respect to P , as well as by all vector fields Hamiltonian with respect to Q. Mathematics Subject Classification. 37K10, 53D17.
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