Fernandes, Rui Loja Connections in Poisson geometry. I: Holonomy and invariants. (English) Zbl 1036.53060 J. Differ. Geom. 54, No. 2, 303-365 (2000). The paper defines the notion of contravariant connections on Poisson manifolds, and gives a detailed study of the consequences of this definition. The motivation comes from the fact that for a non-regular Poisson manifold it is in general impossible to find a covariant connection compatible with the Poisson tensor. Employing the general principle of Poisson geometry that in many situations the cotangent bundle of a Poisson manifold plays the role of the tangent one, the author defines the notion of a contravariant connection on a principal bundle over a Poisson manifold.With this definition, the author shows that every Poisson manifold has a linear contravariant connection preserving the Poisson tensor. The corresponding concepts of parallel transport, curvature, geodesics, holonomy, characteristic classes are defined and studied. In particular, the author obtains several results on the stability of symplectic leaves of a Poisson structure, describes the Lie algebra of the holonomy group, establishes the relation with the usual covariant connections, and defines and computes in several important examples Poisson characteristic classes.In the process, several previously defined concepts of Poisson geometry are recovered: the notion of contravariant derivative coincides with the one introduced earlier by Vaisman; the concept of linear holonomy – with the one studied by Ginzburg and Golubev; finally, the first secondary characteristic class turns out to be (up to a constant) the modular class of a Poisson manifold, defined and studied by Weinstein.The author points out that most of the constructions in the paper can be generalized to an arbitrary Lie algebroid. Reviewer: Olga Radko (Los Angeles) Cited in 1 ReviewCited in 45 Documents MSC: 53D17 Poisson manifolds; Poisson groupoids and algebroids 53C05 Connections (general theory) 53C29 Issues of holonomy in differential geometry Keywords:Poisson manifold; contravariant connection; holonomy; characteristic classes × Cite Format Result Cite Review PDF Full Text: DOI arXiv