= Boundary normal coordinates
{title2=$g=dr^2+h(r)$}
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.
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