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The book I am reading (General relativity, Robert M.Wald, page 12) says that "space-time itself does not (as far as we know), naturally live in a higher dimensional Euclidian space."

My question is how is it possible for an m-dimensional manifold not to lie inside a higher dimensional Euclidian space? How can I, for example, construct a $1$-manifold that is not contained in $\mathbb{R}^2$? This seems very counter-intuitive.

Since an Euclidian space is just characterized by its metric or dot product (or so I assume), then can we say that any manifold for which the metric does not hold does not live inside said Euclidian space?

Ash
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    You can do it (Nash embedding theorem, https://mathoverflow.net/questions/127734/nash-embedding-theorems-for-pseudo-riemannian-manifolds). But the important word here is that there's no “natural” way of doing it. – Hans Lundmark Aug 18 '17 at 07:29
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    It may live in a higher dimension Euclidean space, mathematically. But physically (as far as we know) there is no reason to believe that this is the case. And you can define a $1$-manifold which doesn't live in $\Bbb R^2$ (or any higher dimension). Take the interval $[0,1]$, all by itself, and identify the two end points. That's a circle that doesn't live in a bigger Euclidean space. – Arthur Aug 18 '17 at 07:33
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    Thanks for your comments. I'll check the Nash theorem out. @Arthur, what I don't understant is precisely this: the circle you define can be embedded in $\mathbb{R}^2$... So in what sense doesnt't it live in that space? – Ash Aug 18 '17 at 07:40
  • So, this means that "embedding" is not the same as "lives in"? – Ash Aug 18 '17 at 07:42
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    It doesn't live in that space, because the way I've defined it it doesn't. Of course, you can embed it if you like, and some calculations get massively simplified by doing so, but I haven't embedded it there, and it doesn't live there until I do. And no, "Embedded in" and "lives in" are the same thing (one has a strict definition, the other encapsulates the intuition behind that definition). The verb "Embed" doesn't have a nice corresponding synonym, though, so I use that. – Arthur Aug 18 '17 at 07:42
  • @HansLundmark I quickly read the theorem, so... Why is it considered non natural? – Ash Aug 18 '17 at 07:48
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    Because it's arbitrary! There may be infinitely many ways of doing it, which one is the “right” one? – Hans Lundmark Aug 18 '17 at 07:50
  • @HansLundmark Ah. This makes it clearer. – Ash Aug 18 '17 at 07:51

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