Formal equivalence of Poisson structures around Poisson submanifolds

Details Formal equivalence of Poisson structures around Poisson submanifolds Formal equivalence of Poisson structures around Poisson submanifolds

2010-11-27

arxiv.org/abs/1011.5998v1

Mathematics

Differential Geometry

16 pages

Scientific paper

Let (M, {\pi} ) be a Poisson manifold. A Poisson submanifold $P \in M$ gives rise to an algebroid $AP \rightarrow P$, to which we associate certain chomology groups which control formal deformations of {\pi} around P . Assuming that these groups vanish, we prove that {\pi} is formally rigid around P , i.e. any other Poisson structure on M , with the same first order jet along P as {\pi} is formally Poisson diffeomorphic to {\pi} . When P is a symplectic leaf, we find a list of criteria which imply that these cohomological obstructions vanish. In particular we obtain a formal version of the normal form theorem for Poisson manifolds around symplectic leaves.

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