zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Groupoides symplectiques. (Symplectic groupoids). (French) Zbl 0668.58017
Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon 2/A, 1-62 (1987).
The paper is devoted to the geometric aspects of the theory of symplectic groupoids and algebroids, which was inspired by certain problems in symplectic mechanics. Roughly speaking, a Lie groupoid Γ, whose manifold of units is denoted by Γ 0 , is a smooth groupoid in the sense of Ehresmann and a Lie algebroid, which is a vector bundle over Γ 0 with a bracket operation, is an “infinitesimal version” of a Lie groupoid. A Lie groupoid Γ endowed with a symplectic 2- form is called a symplectic groupoid, if the graph of its multiplication is a lagrangian submanifold in (-Γ)×Γ×Γ. On the other hand, every Poisson manifold (Γ 0 ,Λ 0 ) induces a Lie algebroid structure on T * Γ 0 Γ 0 and every Lie algebroid of such a type is called a symplectic algebroid. The main result of the paper reads that a Lie algebroid is the algebroid of a local symplectic groupoid if and only it it is a symplectic algebroid. This assertion has several concrete applications in symplectic geometry.
Reviewer: O.Kolář

37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
58H99Pseudogroups, differentiable groupoids and general structures on manifolds