I was recently trying to test something out with the Riemann tensor. I used only 2 dimensions for simplicity sake. As I was deriving the Riemann tensor, I noticed that it looked as if all of the elements were going to come out to be 0 (which they all did). Therefore, this coordinate system is flat space. Then I realized something. I was working in 2D. The very definition of 2D alone essentially means flat planes. Therefore it would make sense for a 2D coordinate system to be flat space. Just to make sure however, I must ask this: Is it possible for a 2 dimensional Riemann tensor to ever have a non zero element, or does the fact that it is 2D mean that there can never be any non zero elements (and therefore no curvature)?