Let $S$ be a compact surface in $\mathbb R^3$. Suppose Gaussian curvature is positive $K>0$ and $q\in S$.

I would like to find a coordinate chart $\phi$ at $q$ such that coordinate vectors $\partial_u\phi,\partial_v\phi$ are principal directions of second fundamental form in $q$.

Question: How do I assure it's existence?

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    $\begingroup$ I believe that you will find the relevant material in this question and the answer to it. $\endgroup$ Jul 15, 2018 at 10:02


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