In order to prove "If a mapping is both geodesic and conformal, it is necessarily an isometry" my book assume the metric in geodesic form can be written as $ds^2=du^2+G(u,v)dv^2\tag1$ without justify anything.
The only information the book have is, The equation of geodesics can be written for a surface, $\frac{d^2v}{d^2u}=\Gamma_{22}^1\left(\frac{dv}{du}\right)^3+\left(2\Gamma_{12}^1-\Gamma_{22}^2\right)\left(\frac{dv}{du}\right)^2+\left(\Gamma_{11}^1-2\Gamma_{12}^2\right)\frac{dv}{du}-\Gamma_{11}^2\tag2$
For another place they assume, "If a surface is developable, then there must be a relation between $\frac{df}{dx}$ and $\frac{df}{dy}$", $\frac{df}{dx}\left(\frac{df}{dy}\right)\tag3$
I couldn't come up with their intuition or derivation by self-try. If the surface was given explicitly then maybe I can calculate the metric by $\langle \partial_i,\partial_j\rangle$ but this is not the case. Can anyone help me to figure out to understand $(1)$ and $(3)$?
Any solution or hint will be appreciated.