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Poisson geometry. (English) Zbl 0930.37032

This is a state-of-the-art survey, written by one of the leaders of the field.

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.

Completeness issues are discussed in section 6, with conjectures about the interplay between completeness and the geometry of the symplectic leaves.

Poisson groupoids and their actions are reviewed in sections 7, 8; modular groupoids theory in sections 9, 10.

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.

37J05Relations of dynamical systems with symplectic geometry and topology
37-02Research exposition (Dynamical systems and ergodic theory)
37J60Nonholonomic dynamical systems
58H05Pseudogroups and differentiable groupoids on manifolds
53D17Poisson manifolds; Poisson groupoids and algebroids
53C30Homogeneous manifolds (differential geometry)