Let $\mathbb{T}^n$ be the $n$-dimensional torus, $U \subset \mathbb{T}^n$ an open set and $\phi_1, \phi_2 : U \to \mathbb{R}^n$ local coordinates on $U$ which are inverses of $\pi$, where $\pi : \mathbb{R}^n \to \mathbb{T}^n$ is the projection that takes a point $x = (x_1, ..., x_n) \in \mathbb{R}^n$ to the point in the torus by taking each component modulo one, i.e $$\pi(x_1, ..., x_n) = (x_1, ..., x_n) \mod 1.$$ I don't understand who is the inverse of $\pi$, i.e I don't understand what form have the functions $\phi_1, \phi_2$.
Can someone help me, please?
The torus $\mathbb{T}^n$ is the quotient of $\mathbb{R}^n$ by $f_{p_1,...,p_n}(x_1,...,x_n)=(x_1+p_1,...,x_n+p_n)$ where $p_1,...,p_n$ are integers. Let $\pi:\mathbb{R}^n\rightarrow\mathbb{T}^n$ be the quotient map. If you take $V$ an pen subset of $\mathbb{R}^n$ such that for every $(p_1,...,p_n)$ $f_{p_1,...,p_n}(V)\cap V$ is empty, for example, take $V=B(x,r), r<1$, then $g(V)=U$ is an open subset of $\mathbb{T}^n$ such that ${\pi}_{\mid V}:V\rightarrow U$ is an homeomorphism. Take $\phi_U=({\pi_{\mid V}})^{-1}:U\rightarrow V\subset\mathbb{R}^n$. You can find a covering $(U_i)_{i\in I}$ of $\mathbb{R}^n$ such that $V_i=B(x_i,r_i), r_i<1$. Write $U_i=p(V_i)$, and $\phi_i=(\pi_{V_i})^{-1}$ you obtain an atlas $(U_i,\phi_i)$ of $\mathbb{T}^n$.