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Closed ball is manifold with boundary

  1. Mar 22, 2014 #1
    I've been trying to prove that the closed unit ball is a manifold with boudnary, using the stereographic projection but I cannot seem to be able to get any progress. Can anyone give me a hint on how to prove it? Thanks in advance :)
     
  2. jcsd
  3. Mar 22, 2014 #2

    micromass

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    This is a homework-style problem. But since this is grad mathematics, I'll allow it here. I do ask of the people who answer not to give complete answers, like with any homework problem.

    Anyway, consider the closed unit ball ##B## of ##\mathbb{R}^n##. Consider the ##n##-sphere ##S^n##. Consider the stereographic projection ##\sigma:S^n\rightarrow \mathbb{R}^n##. Can you give a formula for the stereographic projection and its inverse? Can you verify it is continuous.

    Then, what can you say about ##\sigma^{-1}(B)##. It is a part of the ##n##-sphere ##S^n##, but can you describe which part it is?

    Charts can then be found by composing ##\sigma^{-1}## with the projections ##\pi:\mathbb{R}^{n+1}\rightarrow \mathbb{R}^n## which leave some coordinate. Can you show that ##\pi\circ \sigma^{-1}## are continuous? Can you describe these things explicitely? Can you tell what the right codomain is? Etc.
     
  4. Mar 22, 2014 #3
    Thanks! I was stuck in σ^−1(B) for no reason. xD
     
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