Differentiable structure on torus

Question

I defined torus as quotient space of $\mathbb{R}^2$, and $\pi$ is a quotient map. Then for each $ v\in\mathbb{R}^2$ I took a neighborhood $U(v)=B_{1/3}(v)$ and looked on $\pi|U:U\to \pi(U)=V$ and proved that they are bijections. Now for every such $V$ we regard $\pi^{-1}$ as coordinate and I need to compute change of coordinates and prove that in such way torus is equipped with the structure of a differentiable manifold. How should I do it?

Answer 1

What you have to do is to take two different neighbourhoods as the ones you defined, $U_i$ and $U_j$, and check that the following map $$\pi_i^{-1} \circ \pi_j:V_i \cap V_j \longrightarrow V_j \cap V_i$$ is a smooth map. Observe that this map goes from an open set in $\mathbb{R^3}$ to an open set in $\mathbb{R^3}$ (it is actually the same), so proving its smoothness is a matter of smoothness within the euclidean space, so you may use its tools, namely the Implicit Function Theorem and such.

By the way, remember to check that the open sets you defined cover the whole torus; else way you don't get an atlas and then there is no differentiable structure to be proven.