Closed ball is manifold with boundary

[kostas230]

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 :)

 

[micromass]

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.

 

[kostas230]

Thanks! I was stuck in σ^−1(B) for no reason. xD