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If an infinite cylinder is given as $\mathbb{R}\times S^1$, how is $S^1$ defined?

HelloGoodbye
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    $S^1$ is the unit circle – J. W. Tanner Apr 15 '21 at 21:20
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    Why was this question down-voted? Am I not allowed to ask questions like this here, or what? – HelloGoodbye Apr 15 '21 at 21:44
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    I didn't down-vote your question but without knowing anything about you it can be interpreted as you not trying to look it up. It could also be the case that you knew it was a circle but wanted a coordinate-independent definition. There is no way to tell. A good question is usually less terse and shows that you couldn't find the answer through conventional means. – John Douma Apr 15 '21 at 21:48
  • @JohnDouma Okay, I get your point, and thank you for explaining. But like, how do you look up what "S" means in this context? How do you look up unknown mathematical notation in general? So should I have included some google searches to show that none of them yields any useful results; is that really necessary? I figured a lot of people here will for sure know the answer and can just tell me. A simple answer like "$S^1$ is the unit circle, ${(x,y)\in\mathbb{R}^2|x^2+y^2=1}$," really suffices and does not take long time to write. – HelloGoodbye Apr 15 '21 at 22:40
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    @JohnDouma Sorry for my rant, but sometimes I feel like people on SE expect you to write the question more complicated than it really needs to be, or down-vote your question without giving even a hint of why, which just leaves you scratching your head and feeling unwelcome. – HelloGoodbye Apr 15 '21 at 22:46
  • @J.W.Tanner And thank you! :) – HelloGoodbye Apr 15 '21 at 22:47

1 Answers1

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As mentioned in the comments $S^1$ is the $1$-sphere, which is a fancy name for a circle. It is defined differently in different books. Some popular choices are:

  • $\{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \}$
  • $\{ e^{i \theta} \mid \theta \in \mathbb{R} \}$
  • $\mathbb{R} / \mathbb{Z}$
  • $[0,2\pi] \big / 0 \sim 2\pi$

Again, depending on the book (and what kind of geometry you're doing), you can think of this is a topological space, a manifold, a lie group, etc.

I hope this helps ^_^

Chris Grossack
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    Probably more common is $e^{i\theta}$ for $\theta\in[0,2\pi)$ but it's really the same – wormram Apr 15 '21 at 21:25
  • Thanks; yes, this helped! The first two are notations I understand, but the last two, could you perhaps explain how they work? I haven't seen a set divided by something before, be it a number or another set, and I don't understand the "$\sim 2\pi$" notation. – HelloGoodbye Apr 15 '21 at 21:43
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    Both are notions of quotient spaces. The idea is to take the set above the $\big /$ and "glue together" the points below. So $[0,2\pi/ \big / 0 \sim 2\pi$ is saying to take $[0,2\pi]$ and glue the endpoints together. Of course, this gives you a circle. The $\mathbb{R}/\mathbb{Z}$ example is a little bit subtler, but you might want to look here, here, and here for more info. – Chris Grossack Apr 15 '21 at 22:39