How can I exploit this definition to proof that spherical surfaces aren't flat?

Question

I read in a book that a space is flat if it is possible write infinitesimal displacement (squared) as $ds^2=\xi_1 (du^1)^2+\cdots+\xi_n (du^n)^2$ where $\xi_i = \pm 1$: it is flat if it exists a coordinate system that do that. In another book a read that cylindrical surfaces are flat while spherical ones are not. Where is the link between these two claims? I tried starting by (here $R$ is fixed) $$ ds^2 = R^2 \, d\phi^2 + dz^2 \qquad \textrm{Cylindrical case} $$ $$ ds^2 = R^2 \, d \theta^2 + R^2 \sin^2 \theta \, d \phi^2 \qquad \textrm{Spherical case} $$ and then noting that choosing $u^1 = R \phi$ and $u^2 = z$ we get $ds^2 = (du^1)^2+(du^2)^2$. Apparently it is impossible do the same in spherical case. But how can I prove that I can't find $u^1 = u^1 (\theta,\phi)$ and $u^2 = u^2 (\theta,\phi)$ such that $R^2 d \theta^2 + R^2 \sin^2 \theta \, d \phi^2 = (du^1)^2+(du^2)^2 $? It look like I should prove that doesn't exist $u^1 = u^1 (\theta,\phi)$ and $u^2 = u^2 (\theta,\phi)$ such that $$ \left\{ \begin{array}{l} \displaystyle \left( \frac{ \partial u^1 }{ \partial \theta } \right)^2 + \left( \frac{ \partial u^2 }{ \partial \theta } \right)^2 = R^2 \\ \displaystyle \left( \frac{ \partial u^1 }{ \partial \phi } \right)^2 + \left( \frac{ \partial u^2 }{ \partial \phi } \right)^2 = R^2 \sin^2 \theta \\ \displaystyle \frac{ \partial u^1 }{ \partial \theta } \frac{ \partial u^1 }{ \partial \phi } + \frac{ \partial u^2 }{ \partial \theta } \frac{ \partial u^2 }{ \partial \phi } = 0 \end{array} \right. $$ But I have no idea of how to prove that (for some reason these three conditions are not compatible?) and so showing that starting definition of flat space can exploited to prove that cylindrical surface is flat (I did it!) while spherical it is not (I can't do it!).

Answer 1

A good way to solve a problem like this is to compute a quantity that is defined in terms of the metric (i.e. in terms of $ds^2$) and is invariant under coordinate reparametrisations. If the value this quantity takes for the sphere metric is different from the value this quantity takes for a flat 2D metric, then that would prove that there is no coordinate reparametrisation that turns the sphere metric into the flat Euclidean metric.

The Ricci scalar is such a quantity. The Ricci scalar is defined in terms of the metric and is invariant under coordinate reparametrisations.

On a 2-sphere of unit radius, the Ricci scalar evaluates to $2$ everywhere (see here). On flat 2D Euclidean space, the Ricci scalar evaluates to $0$ everywhere. So it is impossible for there to exist a coordinate reparametrisation that converts the metric of the 2-sphere into the metric of flat 2D Euclidean space.