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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!2012-04-05
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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]$2012-04-05
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    There you have it. And how far have you gotten showing that it *is* a diffeo?2012-04-05
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    I was trying to show that $f$ in local coordinates is diffeo2012-04-05
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    Methinks there are some scalings by $2\pi$ missing.2012-04-05
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    @Jr., can you prove it is at least an homeo?2012-04-05
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    I proved that is a bijection, but I dont know why it is homeo2012-04-05
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    Can you write down an inverse?2012-04-05
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    @DanPetersen yes2012-04-05
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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!)2012-04-05

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