2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65637Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metric and the Poisson structure that describes the noncommutativity. I illustrate this by computing the obstructions for well known examples of noncommutative geometries and quantum groups. These rigid conditions may cast doubt on the idea of noncommutatively deformed space-time.27 pages. Serious typo corrected from v3Quantum AlgebraGeneral Relativity and Quantum CosmologyHigh Energy Physics - TheoryDifferential Geometry58B34; 46L65; 53D17text