Boundary normal coordinates ($g=d{r}^2+h(r)$)

The map $(x,r)\mapsto {\exp }_x(r{\nu }_{\mathrm{in}})$ gives a collar in which the metric has this form. Two metrics with the same induced boundary metric can be identified by their normal collars, fixing the boundary and matching full tangent-space metrics there. To extend the local identification globally, interpolate the metrics, form their smoothly varying collars, and extend the resulting collar isotopy velocity by a cutoff. The time-dependent vector field vanishes on the boundary and agrees with that velocity near it, so its global flow gives the desired boundary-fixing diffeomorphism.