Recently I have seen an interesting answer to an "obvious" question. That is "why can we pull a curve back into a line"? And the answer is "because a manifold of dimension one has no curvature". So I was wandering whether that answer is correct or not. Besides, I am not sure why a manifold of dim 1 has no curvature? Looking for someone's answer.
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What do you mean by pulling a curve back into a line? If a curve intersects itself, then there can't be a continuous map from the curve to a line, let alone a homeomorphism or diffeomorphism. – Oct 22 '18 at 16:10
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The curves normally have extrinsic curvatures.
For example a circle with radius $R$ has extrinsic curvature of $1/R$.
May be the answer is that curves do not have intrinsic curvature which is another way of saying the intrinsic curvature of a curve is $0$.
Mohammad Riazi-Kermani
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The curvature of a circle is given by the way you consider it in a plane. But a circle (minus a point) is isometric to an interval. – mfl Oct 22 '18 at 16:02
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That is correct. I think a circle without a point still has curvature even if it is isometric to an interval. – Mohammad Riazi-Kermani Oct 22 '18 at 16:04
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It has extrinsic curvature. But I think that the OP asks for intrinsic curvature. Maybe I am wrong. – mfl Oct 22 '18 at 16:05
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I guess you are correct. It is quite possible for a curve to not have an intrinsic curvature. – Mohammad Riazi-Kermani Oct 22 '18 at 16:14