Let me start with a longish quote from Einstein http://www.bartleby.com/173/24.html In the past, I have characterized this as a matter of "coordinates", but upon thinking about this, that's not quite right - the word choice is incorrect. (I think I'm expressing myself better this time around, but there may still be room for technical improvement). It is really upon introducing a metric (and not coordinates) that the geometry of the disk becomes non-Euclidean. We can assign coordinates to the marble surface however we like, and a zero Riemann tensor will remain zero. When we change the metric, however (our defintion of distance), we _can_ turn a flat marble surface into a non-flat one.