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Define a map $\mathbb{S}^n \to \mathbb{RP}^n$ given by $x \mapsto \{ x,-x\}.$ Clearly this is not a diffeomorphism, but how can one show that it is an immersion and submersion?

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This is a local diffeomorphism, and the result follows.

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yes, by definition I think immersion & submersion means local diffeomorphism, but how do you prove it is local diffeom? –  jj_p Jun 10 '13 at 20:02
    
@Nicolo' Note that $S^n_+=\{x\in S^n, x_0>0\}$ can be the local charts of both $S^n$ and $RP^n$, what is the antipodal map in this chart? –  Ma Ming Jun 11 '13 at 10:06
    
it is the identity, so we get a local diffeomorphism, right? –  jj_p Jun 11 '13 at 13:52
    
@Nicolo' You got it. –  Ma Ming Jun 11 '13 at 14:00

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