Regular parametrization of a surface is conformal iff it preserves angles.

Question

Can anyone give me some hints of how to start the proof, because I have no idea where to start.

I know if a parametrization is conformal, then $E=G$ and $F=0$, where E,F,G are values in the first fundamental form $I=Edu^2+2Fdudv+Gdv^2$.

Answer 1

The cosine of the angle between two vectors $u,v$ with respect to an inner product $\cdot$ is $$\frac{u \cdot v}{\sqrt{(u\cdot u) (v\cdot v)}}.$$ In the parameter domain you have the standard dot product, and in the surface geometry you have $u \cdot v=u^T I v$. Show that when $I$ has the form you are assuming, these two inner products produce the same angle cosine.