  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.
