I am not talking about local flatness. Local flatness means an entire chart can be mapped to a flat euclidean space. For this reason for example a sphere is not flat, because any chart will still be curved. However a sphere does appear flat at the limit of the sphere size going to infinity. If you draw a small triangle on the surface of the earth the angles will approximately add to 180 degrees. As the triangle gets smaller the angles will approach a sum of 180.
Is there any way to make this notion precise? Do there exist riemmanian metrics where this is not true?