Abstract: We are given a compact domain with boundary (e. g. a ball) with an unknown conformal Euclidean metric. The problem is to reconstruct the metric from knowledge of lengths of all closed geodesics in the domain. This problem was formulated more than hundred years ago in geophysics, where the length of a geodesic is called travel-time. An explicit reconstruction of an isotropic velocity field from travel-times was found by G. Herglotz for the spheric Earth model with velocity depending only on depth.

Much later, in the seventies progress in theoretical geophysics (Novosibirsk group, Bernstein, Gerver, Beylkin) gave rise a problem of Riemannian geometry: in which extent a Riemannian metric on a manifold with boundary can be determined from its travel-time function? Finally an answer was found for metrics within a given conformal class.

If the conformal class is not known then a unique determination of a metric is not possible because of shortage of information. A natural question is whether a Riemannian metric in a manifold is rigid with respect to lengths of all closed geodesics (Michel, Gromov, Pestov, Uhlmann,...). The rigidity problem is still far from a complete solution.

More tags: Santalo's volume formula, "Goursat theorem" for a geodesic geometry, integral geometry.

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