I am looking at the following exercise:
Show that a curve $\gamma (t) = \sigma (u(t), v(t))$ on a surface patch $\sigma$ is a line of curvature if and only if
$$(EM − FL) \dot u^2 + (EN − GL) \dot u \dot v + (FN − GM ) \dot v^2=0$$
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We have that $\gamma$ is a line of curvature if the tangent vector of $\gamma$ is a principal vector of $S$ at all points of $\gamma$, so if $W(\dot\gamma)=-\kappa \dot\gamma$, which is equivalent to $\dot {\textbf{N}}=-\kappa\dot\gamma$.
It holds that $\dot\gamma=\sigma_u\dot u+\sigma_v\dot v$.
We have that $E=\|\sigma_u\|^2, \ F=\sigma_u \cdot \sigma_v , \ G=\|\sigma_v\|^2, \ L=\sigma_{uu}\cdot \textbf{N}, \ M=\sigma_{uv}\cdot \textbf{N}, \ N=\sigma_{vv}\cdot \textbf{N}$.
But how do we get the terms $E, \ F, \ G, \ L, \ M, \ N$ ?
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EDIT:
Now an other question of the exericse is the following:
I have done the following:
Suppose that (i) holds, i.e., the second fundamental form of $\sigma$ is proportional to its first fundamental form. Then $$Edu^2+2Fdudv+Gdv^2=\lambda (Ldu^2+2Mdudv+Ndv^2), \ \text{ for some smooth function } \lambda (u,v)$$ so $$E=\lambda L, \ F=\lambda M, \ G=\lambda N$$
Therefore, $$(EM − FL) \dot u^2 + (EN − GL) \dot u \dot v + (FN − GM ) \dot v^2=(\lambda LM − \lambda ML) \dot u^2 + (\lambda LN − \lambda NL) \dot u \dot v + (\lambda MN − \lambda NM ) \dot v^2=0$$ So, we conclude that all parameter curves are lines of curvature.
Is this correct?
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Suppose that (ii) holds, i.e., $F=M=0$.
Then, we have that the matrices $\mathcal{F}_I=\begin{pmatrix} E & F \\ F & G \end{pmatrix}=\begin{pmatrix} E & 0 \\ 0 & G \end{pmatrix}$ and $\mathcal{F}_{II}=\begin{pmatrix} L & M \\ M & N \end{pmatrix}=\begin{pmatrix} L & 0 \\ 0 & M \end{pmatrix}$ are diagonal.
Therefore the matrix of the Weingarten map, $\mathcal{F}_I^{-1}\mathcal{F}_{II}$ is diagonal. The principal curvatures are the eigenvalues of this matrix, which are the elements of the diagonal of the Weingarten map. Then the eigenvectors are $(1,0)$ and $(0,1)$. Does this imply that the tangent vector of the curve is a principal vector of the surface?
P.S. Suppose we have $W(t_1) = \kappa_1t_1, \ W(t_2) = \kappa_2t_2$, then $\kappa_1$ and $\kappa_2$ are called the principal curvatures of $S$, and $t_1$ and $t_2$ are called principal vectors corresponding to $\kappa_1$ and $\kappa_2$.
