  Using 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.
