So if I pick any 2 points on a 2d manifold, say(adsbygoogle = window.adsbygoogle || []).push({}); x1andx2, then the distance between these two points is a secant line that passes through 3 space that isn't part of the manifold. So no matter what, there doesn't exist an point epsilon,e, where ||e||>||0|| and ||x2-x1||<||e||

No matter how small we shrink the neighborhood by decreasing the length ofe, the distance, ||x2-x1|| is the distance of a secant line that does not lie on the curved surface itself. So it seems that there is no neighborhood on a surface that is euclidian.

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# I How is a manifold locally Euclidean?

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