TITLE: First fundamental form and conformal mapping
QUESTION [0 upvotes]: Pressley says that a diffeomorphism is conformal iff the first fundamental forms of two surfaces are proportional and he writes
$E_2du^2+2F_2dudv+G_2dv^2=\lambda(E_1du^2+2F_1dudv+G_1dv^2)$ and then in the expression of angle between two curves in the second surface he substitutes $E_2=\lambda E_1$ $F_2=\lambda F_1$ and $G_2=\lambda G_1$. How to deduce this from the former equation?

REPLY [1 votes]: This is linear algebra: the coefficient of ${\rm d}u^2$ on the left is $E_2$, on the right it is $\lambda E_1$. Same for the others, looking at the coefficients of ${\rm d}u\,{\rm d}v$ and ${\rm d}v^2$.
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