Coordinate patches attached to a 2-manifold In Road to reality page-185, the following photo is shown:



I am a bit confused here because I thought the charts of the manifold existed as flat sheets which are subsets of $R^2$ (for a 2- manifold of course). So what are these curved coordinate patches which Penrose has shown here?
 A: These coordinate patches as seen in the picture are open sets $U_j\subset M$ subordinate to the atlas $(\varphi_i,U_j)$ of the manifold $M$ equipped with a coordinate grid that is induced by the corresponding chart $\varphi_j$ in the following way:
Let $p\in M$ be any point within the manifold and $U$ be a neighborhood of $p$ with corresponding chart $\varphi$ in the atlas and let $\tilde{U}\subset \mathbb{R}^2$ be the set that is mapped homeomorphically to $U$ by $\varphi$, i.e. $\varphi:\tilde{U}\overset{\sim}{\rightarrow}U$. Taking the canonical grid of $\mathbb{R}^2$, $$\Gamma:=\left\{(x,y):\ x\in\mathbb{Z}\ \vee\ y\in\mathbb{Z}\right\}\, ,$$
and restricting it to $\tilde{U}$, we can map the canonical grid onto the manifold, which is nothing but the image
$$ \Gamma_{\varphi}:=\varphi\left(\tilde{U}\cap\Gamma\right)$$
of the grid under $\varphi$, wherever this is allowed.
These grids heavily depend on the choice of chart $\varphi$ and thus show us, how different choices of coordinates, which one can denote by $(x,y)=\varphi^{-1}$, impact the perception of the point chosen and of those nearby in $\mathbb{R}^2$, when working in that chart/set of coordinates.
