Existence of coordinate charts which coordinate vectors are principal directions

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



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



Question: How do I assure it's existence?




geometry differential-geometry riemannian-geometry




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    I believe that you will find the relevant material in this question and the answer to it.
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up vote
2
down vote

favorite
2












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



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



Question: How do I assure it's existence?




geometry differential-geometry riemannian-geometry




share|cite|improve this question















  • 1




    I believe that you will find the relevant material in this question and the answer to it.
    – Yuri Vyatkin
    Jul 15 at 10:02












up vote
2
down vote

favorite
2









up vote
2
down vote

favorite
2






2





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



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



Question: How do I assure it's existence?




geometry differential-geometry riemannian-geometry




share|cite|improve this question











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



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



Question: How do I assure it's existence?





geometry differential-geometry riemannian-geometry





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 14 at 19:08









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  • 1




    I believe that you will find the relevant material in this question and the answer to it.
    – Yuri Vyatkin
    Jul 15 at 10:02












  • 1




    I believe that you will find the relevant material in this question and the answer to it.
    – Yuri Vyatkin
    Jul 15 at 10:02







1




1




I believe that you will find the relevant material in this question and the answer to it.
– Yuri Vyatkin
Jul 15 at 10:02




I believe that you will find the relevant material in this question and the answer to it.
– Yuri Vyatkin
Jul 15 at 10:02















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