×

Suites d’Atiyah, feuilletages et quantification géométrique. (Atiyah sequences, foliations and geometric quantification). (French) Zbl 0596.57017

Sémin. Géom. Différ., Univ. Sci. Tech. Languedoc 1984/1985, 39-59 (1985).
Transitive Lie algebroids (in the sense of J. Pradines [C. R. Acad. Sci., Paris, Sér. A 264, 245-248 (1967; Zbl 0154.217)] are associated to transversely complete (in the sense of P. Molino [Ann. Sci. Ec. Norm. Supér., IV. Sér. 10, 289-307 (1977; Zbl 0368.57007)]) foliations and to symplectic manifolds. The existence of their principal realizations is studied. Also, some obstruction for the existence of a principal realization of an arbitrary transitive Lie algebroid is defined. This part is related to the paper by K. Mackenzie [Cohomology of locally trivial groupoids and Lie algebroids, Thesis, Monash Univ. (1979); see also Bull. Aust. Math. Soc. 20, 475-477 (1979)].
Reviewer: P.Walczak

MSC:

57R30 Foliations in differential topology; geometric theory
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
55S35 Obstruction theory in algebraic topology
17B99 Lie algebras and Lie superalgebras