  We introduce a non-commutative product for curved spacetimes, that can be
regarded as a generalization of the Rieffel (or Moyal-Weyl) product. This
product employs the exponential map and a Poisson tensor, and the deformed
product maintains associativity under the condition that the Poisson tensor
$\Theta$ satisfies $\Theta^{\mu\nu}\nabla_{\nu}\Theta^{\rho\sigma}=0$, in
relation to a Levi-Cevita connection. We proceed to solve the associativity
condition for various physical spacetimes, uncovering non-commutative
structures with compelling properties.
