Noncommutative Einstein spaces and TQFT

We have studied categorical properties of the noncommutative counter parts of so-called Einstein spaces (their Ricci tensor is proportional to the metric) in the framework of twisted gravity. We have computed the deformed Riemannian tensor and scalar curvature in the formalism of twisted gravity too. We could already see for some examples the remarkable property that being an Einstein space seems to be stable under deformation, using Killing vector field in the twist. The deformed Levi-Civita connection and the deformed Riemann tensor are just the undeformed ones. As a generalization, one should study star geometries, where the noncommutative vector fields are not Killing vectors. On the other hand, the main result of this talk can be summarized as a construction of extended topological quantum field theory.