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?
Differentiable structure on torus
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differential-geometry
mathematical-physics