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Any regular curve may be parametrized by the arc length (the natural parametrization or the unit speed parametrization).

But I haven't seen an analogous development for regular parametric surfaces. I hope we can do this at least for orientable surfaces with no umbilical points. For such a surface, there will be two orthogonal lines of curvature through each point, and I suspect that there is a parametrization whose parametric curves coincide with lines of curvature. But I'm not sure how I should approach showing the existence of such a re-parametrization.

  • Is there a such parametrization?
  • If not, what are the conditions that we should impose to have such a parametrization?
janmarqz
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Bumblebee
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  • Even better, every (sufficiently smooth) has “isothermal parameters” — i.e., a parametrization where the first fundamental form has $E=G$ and $F=0$. But reparametrizing explicitly is harder even than in the case of curves. – Ted Shifrin Jul 13 '23 at 00:17
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    Yes, again, in principle, away from umbilics you can parametrize so that the parameter curves are lines of curvature. See DoCarmo or p. 119-120 of my diff geo text, linked in my profile, – Ted Shifrin Jul 13 '23 at 00:25
  • @TedShifrin: Your comment provides exactly the answer I was looking for "isothermal parameters". If you can write this as an answer, I am more than happy to accept and this question will not be in the unanswered line anymore. – Bumblebee Jul 27 '23 at 13:01

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As I commented, every (sufficiently smooth) surface admits isothermal parameters — i.e., a parametrization in which the first fundamental form has $E=G$ and $F=0$. (That is, it is conformally flat.) An explicit such reparametrization is typically much harder than the parametrization of curves by arclength.

Ted Shifrin
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