On the local structure of noncommutative deformations

Let (M,π,D)(M,π,D) be a Poisson manifold endowed with a flat, torsion-free contravariant connection. We show that if DD is an FF-connection then there exists a tensor T such that DT is the metacurvature tensor introduced by E. Hawkins in his work on noncommutative deformations. We compute T and the metacurvature tensor in this case and show that if T=0 then near any regular point ππ and DD are defined in a natural way by a Lie algebra action and a solution of the classical Yang–Baxter equation. Moreover, when DD is the contravariant Levi-Civita connection associated to ππ and a Riemannian metric, the Lie algebra action can be chosen in such a way that it preserves the metric. This solves the inverse problem of a result of the second author.