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Ok, I think it's all starting to make sense a bit more now.Now once you define the metric tensor and the Levi-Civita connection you can ask wether a particular choice of coordinates is cartesian or not. If the Levi-Civita connection induced by the metric is curved then you cant have cartesian coordinates except for a very small patch on the manifold.

Just to clarify though (and then I'll stop bugging everyone), if we consider a manifold with a metric that induces a non-Euclidean geometry, then if we consider a patch on such a manifold that is large enough that the local geometry cannot be considered as Euclidean, the coordinate maps that we use to map points in such a patch will be non-Cartesian as it will not be possible to construct such coordinate maps (unless we consider smaller patches around each point in the patch)?