tailieunhanh - Đề tài " Integrability of Lie brackets "

In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications. Contents 0. Introduction | Annals of Mathematics Integrability of Lie brackets By Marius Crainic and Rui Loja Fernandes Annals of Mathematics 157 2003 575 620 Integrability of Lie brackets By Marius Crainic and Rui Loja Fernandes Abstract In this paper we present the solution to a longstanding problem of differential geometry Lie s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive explain and improve the known integrability results we establish integrability by local Lie groupoids we clarify the smoothness of the Poisson sigma-model for Poisson manifolds and we describe other geometrical applications. Contents 0. Introduction 1. A-paths and homotopy . A-paths . A-paths and connections . Homotopy of A-paths . Representations and A-paths 2. The Weinstein groupoid . The groupoid Q A . Homomorphisms . The exponential map 3. Monodromy . Monodromy groups . A second-order monodromy map . Computing the monodromy . Measuring the monodromy The first author was supported in part by NWO and a Miller Research Fellowship. The second author was supported in part by FCT through program POCTI and grant POCTI 1999 MAT 33081. Key words and phrases. Lie algebroid Lie groupoid. 576 MARIUS CRAINIC AND RUI LOJA FERNANDES 4. Obstructions to integrability . The main theorem . The Weinstein groupoid as a leaf space . Proof of the main theorem 5. Examples and applications . Local integrability . Integrability criteria . Tranversally parallelizable foliations Appendix A. Flows . Flows and infinitesimal flows . The infinitesimal flow of a section References 0. Introduction This paper is concerned with the general problem of integrability of geometric structures. The geometric structures we consider are always associated with local Lie brackets on sections of some vector bundles or what one calls Lie algebroids. A Lie algebroid can be thought of as a generalization of the

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