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 $R^n$?