# $S^1 \times S^1 \ldots \times S^1$ is diffeomorphic to $\mathbb{T}^n$

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 $\mathbb{R}^n$

I defined $f$ as in comments, proved that it is a bijection, and stuck here...
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
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
I was trying to show that $f$ in local coordinates is diffeo –  Jr. Apr 5 '12 at 19:16