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how do I show that $S^1 \times S^1 \ldots \times S^1 $ is diffeomorphic to $\mathbb{T}^n$ as manifolds? Where $S^1 \times S^1 \ldots \times S^1 $ has the natural differential structure of a product manifold and $\mathbb{T}^n$ is obtained by the action of the group of integer translation of $\mathbb{R}^n$ in $\mathbb{R}^n$

ADDED(04/05/12):

I defined $f$ as in comments, proved that it is a bijection, and stuck here...

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Can you see a candidate map for the diffeo? Doing the case $n=1$ first should be quite indicative of the general one! –  Mariano Suárez-Alvarez Apr 5 '12 at 19:00
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Surely, $f \colon S^1\times S^1 \ldots \times S^1 \rightarrow \mathbb{T}$ such that $f(e^{it_1},e^{it_2},\ldots , e^{it_n})=[t_1,t_2,...,t_n]$ –  Jr. Apr 5 '12 at 19:08
    
There you have it. And how far have you gotten showing that it is a diffeo? –  Mariano Suárez-Alvarez Apr 5 '12 at 19:09
    
I was trying to show that $f$ in local coordinates is diffeo –  Jr. Apr 5 '12 at 19:16
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Dear Jr., it is better if you write down everything you have done in the body of the question. It allows us to help you exactly at the point you got stuck (and avoids our having to ask 25 questions to find where that point is!) –  Mariano Suárez-Alvarez Apr 5 '12 at 20:12

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