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I have been doing some reading on tori. What I can make out of it is that a torus can be equipped with different metrics -- locally Euclidean or as an embedded surface. It is said however that the torus with the locally Euclidean metric cannot be realized as an embedded surface. Why is this true and what is the metric as an embedded surface like? Why would we want the latter metric, since it seems to me the former is more natural?

Thanks.

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    You have to be careful when you say "embedded metric". It depends on which space you embed it in. The flat torus (the locally Euclidean one) can be isometrically embedded in $\mathbb{R}^4$, but not in $\mathbb{R}^3$.2012-03-09
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    @ThomasRot: Thanks! Why can't it be isometrically embedded in $R^3$? isn't it just a surface/ subset of $R^3$?2012-03-09
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    I was browsing around, and apparently I was wrong. http://mathoverflow.net/questions/31222/c1-isometric-embedding-of-flat-torus-into-mathbbr3 .2012-03-09
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    The standard embedding http://en.wikipedia.org/wiki/Torus however is not flat in $\mathbb{R}^3$. The embedding of the torus as $S^1\times S^1\subset \mathbb{R}^2\times \mathbb{R}^2$ is flat I believe.2012-03-09
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    The torus is compact. If it is smoothly embedded in $\mathbb{R}^3$ it must have at least one point of positive curvature. From this it can be deduced that it cannot be flat with the metric induced by the embedding. Depending on the background knowledge you have this might require some additional research/work to see this.2012-03-09
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    Usually we say *tori* instead of *toruses*.2012-03-11
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    @Thomas: Here's an answer I gave earlier with a complete (but almost certainly improvable) proof of "every closed surface isometrically embedded in $\mathbb{R}^3$ has a point of positive curvature." Consider this the additional research/work for Hilbert. http://math.stackexchange.com/questions/89061/are-there-any-compact-embedded-2-dimensional-surfaces-in-mathbb-r3-that-are/89073#890732012-07-31
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    @JasonDeVito: not completely, but thanks for the link. It also still remains to understand why a surface, when having a point with positive curvature, cannot be flat there, i.e. isometrically embedded into flat two space. It's not that difficult, but requires computation.2012-07-31

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