Let $(M,\pi,\mathcal{ D})$ be a Poisson manifold endowed with a flat, torsion-free contravariant connection. We show that if $\mathcal{ D}$ is an $\mathcal{ F}$-connection then there exists a tensor $\mathbf{ T}$ such that $\mathcal{ D}\mathbf{ T}$ is the metacurvature tensor introduced by E. Hawkins in his work on noncommutative deformations. We compute $\mathbf{ T}$ and the metacurvature tensor in this case, and show that if $\mathbf{ T}=0$ then, near any regular point, $\pi$ and $\mathcal{ D}$ are defined in a natural way by a Lie algebra action and a solution of the classical Yang-Baxter equation. Moreover, when $\mathcal{ D}$ is the contravariant Levi-Civita connection associated to $\pi$ and a Riemannian metric, the Lie algebra action preserves the metric.
Categories: math.DG