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I am reading the beginning of the first volume on the introduction to Differential Geometry by Spivak.

In the first part he defines manifolds and states that 'the only connected 1-manifolds are the line $\mathbb{R}$ and the circle $\mathbb{S}^1$'.

(1) Is that 'only' a direct consequence of the definition of manifold? I do not really see the answer. It is also true that, at this point of the book, Spivak uses results from Algebraic topology/Homology (without proving them) to give a wider outlook on the world of manifolds. So it might well be that the answer is not trivial. If it actually is, I would like to see why.

(2) Does this fact imply that the only surfaces of revolution in $\mathbb{R}^3$ which can be obtained by connected 1-manifolds are cylinders and 2-tori?

Gibbs
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    It is certainly not a direct implication of the definition, and though it is not too hard to prove, it is not trivial either. – Lukas Geyer May 27 '18 at 14:44
  • Can you give an idea or the proof/provide a reference? – Gibbs May 27 '18 at 14:47
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    Here is a "proof," which is not really a proof because it sweeps away a few important ingredients. You're trying to classify curves which do not intersect themselves and do not have endpoints. No matter how wild a curve is, you could parametrize it by arc length. If the arc length parameter is injective, it must be a wiggly curve homeomorphic to a line. Since the curve doesn't intersect itself, if it's not injective, it's periodic, and you have something that looks like a wiggly loop, homeomorphic to a circle. – A. Thomas Yerger May 27 '18 at 14:47
  • @AlfredYerger thanks for the idea. Can you suggest any book where I can find a formalisation of this? – Gibbs May 27 '18 at 14:49
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    Do Carmo's curves and surfaces in its first chapter explains how to parametrize curves by arc length once they are embedded. You need a theorem that says this is always possible. The Whitney theorem is probably overkill, but the easy Whitney theorem shows it can be done in dimension $2(1) + 1 = 3$, which is exactly the case Do Carmo treats in his curves and surfaces book. Easy Whitney appears in Spivak a little later on. – A. Thomas Yerger May 27 '18 at 14:51
  • Possible duplicate of The only 1-manifolds are $\mathbb R$ and $S^1$ – Lee Mosher May 27 '18 at 15:14
  • Here's another duplicate with slightly different point of view https://math.stackexchange.com/questions/1304107/characterization-of-1-dimensional-manifolds?rq=1 – Lee Mosher May 27 '18 at 15:16
  • @LeeMosher I have edited my post with a supplementary sub-question. – Gibbs May 27 '18 at 15:17
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    It's not super trivial to prove this. In fact, "the line with two origins" and "the long line" are near-misses. They are both locally homeomorphic to $\mathbb R$, but the first fails to be Hausdorff and the second fails to be second countable. – Cheerful Parsnip May 27 '18 at 17:02

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You could try the following resources: this or this or (this (all lecture notes for classes). Or on this site as well, with bonus link. This one looks pretty formal.

Henno Brandsma
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