Boundary rigidity

A Riemannian metric is boundary rigid within a specified class if the distances it induces between all pairs of boundary points determine the interior metric, up to a diffeomorphism that fixes the boundary pointwise. The qualification is unavoidable: relabeling interior points in this way changes the coordinate expression of the metric without changing any boundary distance.

This inverse problem idealizes recovering an interior wave-speed geometry from boundary travel times. Its data are shortest-path distances, so arbitrary trapped regions need not be visible. For a Euclidean disk, multiplying the metric by $c^2$ multiplies every boundary distance by $c$; this simple experiment already distinguishes a uniform change of travel-time scale. Determining a general spatially varying metric is much harder. A landmark theorem proves boundary rigidity for compact simple two-dimensional Riemannian manifolds, where the boundary is strictly convex and the relevant geodesics have no conjugate-point ambiguity.^[ref]