Prove that a parametrization $x(u,v)$ is conformal (angle-preserving) if and only if the coefficients of the first fundamental form satisfy $E=G$ and $F = 0$.
My attempt:
It suffices to consider $e_1$ and $e_2$ in $\mathbb{R}^2$. It is easy to show that the differential $dx(e_1)=x_u$ and $dx(e_2)=x_v$. Note that $e_1$ and $e_2$ are orthogonal to each other, while $x_u$ and $x_v$ are also orthogonal to each other in the corresponding tangent space.
Now, note that, by assumption, $$\cos \theta = \frac{\langle e_1,e_2\rangle}{\|e_1\|\cdot \|e_2\|} = \frac{\langle x_u,x_v\rangle}{\|x_u\|\cdot\|x_v\|} = \frac{F}{\sqrt{EG}}.$$
However, $\langle e_1,e_2 \rangle=0$ implies $F=0$.
What about $E$ and $G$? I appreciate any hints here. Thanks in advance.
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Best Answer
Hint: $e_1 +e_2$ and $e_1 - e_2$ are perpendicular too.