The Sphere Does Not Admit a Metric Isometry into a Euclidean Space. – math.stackexchange.com

The Sphere Does Not Admit a Metric Isometry into a Euclidean Space. – math.stackexchange.com

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Problem. There is no embedding $f:S^2\to \mathbf R^3$ such that for all points $p, q\in S^2$, we have $d_{S^2}(p, q)=d_{\mathbf R^3}(f(p), f(q))$. Here $d_{S^2}$ is the metric on $S^2$ which is ...

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