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I have a question about the curvature line parametrization. We said that for a given surface $f: U \rightarrow \mathbb{R}^3$ we find a local curvature line parametrization such that both the first and second fundamental form are diagonal matrices. Although I understand that the shape operator of weingarten operator is immediately diagonal in this coordinate system, I don't understand why we know that the fundamental forms need to be diagonal in this coordinate system. Does anybody know this? Thus, if we parametrize everything with respect to the main curvature directions, why do we get diagonal matrices? (Actually, I would understand it for the first fundamental form, as the principal curvature directions are orthogonal to each other, but I have troubles to see it for the second one).

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This is the definition of lines of curvature, which gives you a diagonal shape operator matrix $S$. But this matrix is the product of the inverse of the first fundamental form matrix $\mathbf I$ with the second fundamental form matrix. Since $\mathbf I$ and $S$ are diagonal, so is their product.

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