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?
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.