Think for example of the torus as a square with the proper edges identified. Viewed like this (i.e. using the flat metric), it clearly has zero curvature everywhere. More specifically, it seems Euclid's axioms are satisfied. But however we have non-trivial topology. So what's up? Or is Euclid's circle axiom not satisfied? But anyway my question in general is: is it clear Euclid's axioms forbid non-trivial topology? (In other words: I see that they imply Euclidean geometry locally, but why globally?) Or is it not necessary?