2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101198Noncommutatively 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.36 pages, 1 figure. Expands upon my earlier paper math.QA/0211203Quantum AlgebraDifferential GeometrySymplectic Geometry58B34; 46L65; 53D17text