TAILIEUCHUNG - Đề tài " Two dimensional compact simple Riemannian manifolds are boundary distance rigid "

We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction. 1. Introduction and statement of the results Let (M, g) be a compact Riemannian manifold with boundary ∂M . Let dg (x, y) denote the geodesic distance between x and y. The inverse problem we address in this paper is whether we can determine the Riemannian metric g knowing dg (x, y) for any x ∈ ∂M , y ∈ ∂M . . | Annals of Mathematics Two dimensional compact simple Riemannian manifolds are boundary distance rigid By LeonidPestov and Gunther Uhlmann Annals of Mathematics 161 2005 1093 1110 Two dimensional compact simple Riemannian manifolds are boundary distance rigid By Leonid Pestov and Gunther Uhlmann Abstract We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional compact simple Riemannian manifold with boundary we can determine uniquely the Riemannian metric up to the natural obstruction. 1. Introduction and statement of the results Let M g be a compact Riemannian manifold with boundary dM. Let dg x y denote the geodesic distance between x and y. The inverse problem we address in this paper is whether we can determine the Riemannian metric g knowing dg x y for any x E dM y E dM. This problem arose in rigidity questions in Riemannian geometry M C Gr . For the case in which M is a bounded domain of Euclidean space and the metric is conformal to the Euclidean one this problem is known as the inverse kinematic problem which arose in geophysics and has a long history see for instance R and the references cited there . The metric g cannot be determined from this information alone. We have dý g dg for any diffeomorphism ý M - M that leaves the boundary pointwise fixed . ý dM Id where Id denotes the identity map and vd g is the pull-back of the metric g. The natural question is whether this is the only obstruction to unique identifiability of the metric. It is easy to see that this is not the case. Namely one can construct a metric g and find a point x0 in M so that dg x0 dM supx yeQMdg x y . For such a metric dg is independent of a change of g in a neighborhood of x0 . The hemisphere of the round sphere is another example. Part of this work was done while the author was visiting MSRI and the University of Washington. Partly supported by NSF and a John Simon Guggenheim Fellowship. 1094 LEONID PESTOV AND GUNTHER UHLMANN Therefore it

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