# The Cauchy Data and the Scattering Relation

## Abstract

As mentioned in the preface to this volume a combination of unique continuation results with the boundary control method has led to the solution of the inverse problem of determining a metric of a Riemannian manifold (with boundary) from the dynamic Dirichlet-to-Neumann map associated with the wave equation. Although these results are very satisfactory it requires too much information. By just looking at the singularities of the dynamic Dirichlet-to-Neumann (DN) map one can determine the boundary distance function (the minimal travel time along geodesies connecting points on the boundary of a Riemannian manifold) in the case that there are no conjugate points of the metric, i. e. no caustics. This is shown in Section 3 of this article using geometrical optics expansions. A natural question to ask if one can determine the metric from this data alone; this question is at the center of the boundary rigidity problem studied in Riemannian geometry which is one of the main topics of this volume.

### Keywords

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