Given an arbitrary parameterization of a differentiable curve $r: R\rightarrow R^n$ in terms of some variable $t$, we can re-parameterize (although perhaps not in closed-form) $r$ in terms of its arclength $s$. This gives a handy representation where as $s$ increases by 1, we move 1 unit along $r$ in $R^n$.

Is is 1) possible and 2) comparably straightforward to do this for surfaces? I.e. if $r: R^m \rightarrow R^n$ is differentiable can we reparameterize $r$ so that moving 1 unit in $R^m$ moves us 1 unit along the surface $r$ in $R^n$?

  • $\begingroup$ I like to think of the Gauss map of an analogue of "arclength reparametrization" for a surface. $\endgroup$ – MSobak Oct 29 '18 at 17:39

No, this latter would mean that any manifold is locally isometric to an Euclidean space with its flat metric. This is only true in dimension $1$ by the mean of arclength parametrization as you pointed out, the first obstruction coming to my mind being the curvature.

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    $\begingroup$ The quantity of things that don't generalize neatly from 1-D to n-D causes me no end of amazement, delight and frustration. $\endgroup$ – Scott Oct 29 '18 at 18:06
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    $\begingroup$ I agree that the delight part is strong here: the article images.math.cnrs.fr/Un-theoreme-et-une-part-de-pizza (in french) explains curvature and how the fundamental theorem dealing with it helps us to properly eat pizzas. $\endgroup$ – Balloon Oct 29 '18 at 18:14
  • $\begingroup$ This is infinitely more clear after having read the pizza article. :) $\endgroup$ – Scott Oct 29 '18 at 20:49
  • $\begingroup$ Glad to hear that! :) $\endgroup$ – Balloon Oct 29 '18 at 20:50

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