I'm having trouble understanding how a space can be locally Euclidean, with zero curvature, and globally non-Euclidean, with curvature. If the space had locally approximately zero curvature, I see how global non-zero curvature could arise. If it is exactly zero, however, how can it accumulate to be non-zero?
My professor said my issue was due to me imagining the local region to be finite in size, whereas it was actually infinitesimal. I understand infinitesimals to be arbitrarily small, but finite, so I don't see how this makes a difference. Is the local geometry only Euclidean in the sense it approaches Euclidean geometry as the size/area/volume of the region approaches 0, but for any finitely sized region, the curvature would be non-zero?
In this case, would it then be true that there is no space with global curvature^ where the curvature is zero for any finitely sized region (and thus every region that is physical)?
^except maybe at any stationary point that existed, if you had an open ball containing only that point?
If my question isn't clear, it's probably due to my inexperience with this field and these terms; please let me know what doesn't make sense and I'll try to rephrase!