zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.


MSC:
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)