  Noncommutatively deformed geometries, such as the noncommutative torus, do
not exist generically. I showed in a previous paper that the existence of such
a deformation implies compatibility conditions between the classical metric and
the Poisson bivector (which characterizes the noncommutativity). Here I present
another necessary condition: the vanishing of a certain rank 5 tensor. In the
case of a compact Riemannian manifold, I use these conditions to prove that the
Poisson bivector can be constructed locally from commuting Killing vectors.
