  We discuss contravariant connections on Poisson manifolds. For vector
bundles, the corresponding operational notion of a contravariant derivative had
been introduced by Izu Vaisman. We show that these connections play an
important role in the study of global properties of Poisson manifolds and we
use them to define Poisson holonomy and new invariants of Poisson manifolds.

Differential Geometry;Symplectic Geometry