Suppose two curves $\gamma$ and $\gamma'$ are diffeomorphic.
Is the arc-length measure $ds_\gamma$ absolutely continuous to $ds_\gamma'$ with a positive derivative? ($ds_\gamma=\phi\, ds_\gamma'$ for some $\phi\geq 0$)
Does the similar result hold for diffeomorphic surfaces in $\mathbb{R}^n$?