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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?

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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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  • $\begingroup$ If that’s the case in a previous theorem about isometry the following expression occurs $E_1du^2+2F_1dudv+G_1dv^2=E_2du^2+2Fdudv+G_2dv^2$. Here to show $E_1=E_2, F_1=F_2,G_1=G_2$ Pressley takes some specific curves. Why not just equate the coefficients like here? $\endgroup$
    – danny
    Apr 12, 2022 at 1:42
  • $\begingroup$ math.stackexchange.com/questions/3546661/… This question. What’s the difference? $\endgroup$
    – danny
    Apr 12, 2022 at 1:58

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