I was reminded of this point by a recent discussion in the "Classification of manifolds ..." thread. The question is this: As I understand it, the NCG framework requires a Riemannian manifold. Given this, at best one could hope to obtain a Euclidean theory, right? So is it fair to say that the NCG framework does not include real dynamics? I note that even in field theory there are many non-trivial issues about going from imaginary to real time (so that it is, as a practical matter, not possible to do so). In quantum gravity the situation is, I think, much worse. Other related questions: In the Riemannian framework, can the NCG program describe actions which are not real (so that the "Boltzmann weight" is not positive)? Are there "non-relativistic" NCG models, where time is conventional but the spatial geometry is treated in an interesting way. I think I know of models which are roughly like this, e.g. quantum Hall physics, but perhaps its well studied in some other context?