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Zbl 0930.37032
Weinstein, Alan
Poisson geometry. (English)
[J] Differ. Geom. Appl. 9, No.1-2, 213-238 (1998). ISSN 0926-2245

This is a state-of-the-art survey, written by one of the leaders of the field. \par First, the local structure is described, with emphasis on linear, quadratic structures and their perturbations. Then the advances in global problems are reviewed, beginning with a general programme in section 4. Homology and cohomology studies, which began with Lichnerowicz in 1977, are an active area of research. \par Completeness issues are discussed in section 6, with conjectures about the interplay between completeness and the geometry of the symplectic leaves. \par Poisson groupoids and their actions are reviewed in sections 7, 8; modular groupoids theory in sections 9, 10. \par In sections 11, 12, the author offers research ideas and open problems (as usual in his papers). A caveat about possible omissions in the literature should not be taken seriously. For instance, even nonholonomic brackets (where the Jacobi identity is not satisfied) are mentioned and encouragement given for their study.
[Jair Koiller (Rio de Janeiro)]

MSC 2000:: *37J05 Relations with symplectic geometry and topology
37-02 Research exposition (Dynamical systems and ergodic theory)
37J60 Nonholonomic dynamical systems
58H05 Pseudogroups on manifolds
53D17 Poisson manifolds
53C30 Homogeneous manifolds

Keywords: Poisson manifold; symplectic manifold; Lie algebroid; Lie groupoid; foliation; Poisson Lie group; homogeneous space; modular automorphism group; phase space

Cited in: Zbl 0992.53061

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