Are there some standard techniques to find charts and atlas of a manifold? I'm looking for an easy way to find charts (can be trivial ones) so that I could easily find more examples to play with and test some properties of manifolds.
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What do you mean by "to find charts and atlas of a manifold"? In order to speak about manifolds you need an atlas to begin with. – freakish Feb 20 '17 at 20:36
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@freakish suppose you already know a particular set is a manifold (you don't know the atlas yet, you know it's a manifold using a theorem for example), is there some techniques to find its atlas? – user42912 Feb 20 '17 at 20:41
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Well, a set is not a manifold. You need this additional structure. If some theorem says "$X$ is a manifold" then it always has some implicit atlas under the hood. You just have to read between lines how it is constructed. For example when someone says "$X\times Y$ is a manifold" then what is hidden from you is that the atlas is defined by taking (cartesian) products over all possible pairs of charts from $X$ and $Y$. Now some constructions can be complicated and you may lose track of those charts. But the only advice I can give you is to carefuly read line by line. – freakish Feb 20 '17 at 21:04
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So as you can see you cannot know that a "set" is a manifold without knowing the atlas. It's just impossible. It's like asking "how can I find how multiplication works in a group $G$?" Well, you have to carefully read how the group is constructed. There are no easy ways. – freakish Feb 20 '17 at 21:09
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@freakish I agree with you. Ok, suppose I know the atlas. If I want to find another, how can I do that? – user42912 Feb 21 '17 at 04:35