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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]

Gunther Uhlmann studies inverse problems, imaging and partial differential equations. His research asks how measurements made outside an inaccessible region constrain its internal coefficients or geometry. His institutional research account identifies applications to medical imaging and the structure of Earth.^[ref]

With Leonid Pestov, he proved a two-dimensional boundary rigidity theorem: for a compact simple Riemannian manifold, boundary distances recover the metric up to a boundary-fixing diffeomorphism. The qualification prevents confusing a recoverable physical geometry with a unique coordinate description.^[ref]

This gives a precise version of the imaging question “can travel times reveal the interior?” It also explains why assumptions about ray geometry matter: if measurements never probe a hidden region, changing that region need not change the data.

Leonid Pestov is a mathematician associated with inverse problems for waves and geometry. With Gunther Uhlmann, he proved boundary rigidity for compact simple two-dimensional Riemannian manifolds: boundary-to-boundary distances determine the interior metric up to a change of coordinates fixing the boundary.^[ref]

The ambiguity is physical as well as mathematical. Relabeling interior points does not change the travel time between labeled boundary points, so no reconstruction using only those times can distinguish the relabeling. The theorem shows that, under its geometric assumptions, this is the entire ambiguity. Pestov's institutional lecture description also concerns reconstructing coefficients of a wave equation from observations, connecting the geometric theorem with travel-time and wave-imaging measurements.^[ref]