A Riemannian manifold is said to be flat if the curvature is 0 everywhere. An example in dimension 1 is the circle. However, I cannot see how the curvature of the circle could be 0. See for instance:
How could the curvature of a circle be 0? How to show the Riemannian curvature is 0?
Please help.
When you look at a circle, you are seeing its extrinsic curvature, which is also what your link is calculating. That is a property of how the circle is imbedded into another manifold, not a property of the circle as a manifold itself.
The curvature being referred to here is the intrinsic curvature, which is defined strictly in terms of the manifold itself, not any imbedding. 1-dimensional manifolds are incapable of supporting any curvature, so the circle is flat.