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Two dimensional compact simple Riemannian manifolds are boundary distance rigid. (English) Zbl 1076.53044
Consider a compact Riemannian manifold \(M\) with boundary \(\partial M\) and denote by \(d_g(x,y)\) the geodesic distance between points \(x,y\in\partial M\). The authors study the question whether one can determine the Riemannian metric \(g\) knowing \(d_g(x,y)\) for any \(x,y\in\partial M\). R. Michel [Invent. Math. 65, 71–83 (1981; Zbl 0471.53030)] conjectured that simple manifolds are boundary distance rigid, that is, \(d_g\) determines \(g\) uniquely up to an isometry which equals the identity on the boundary. In this paper, it is shown that simple two-dimensional compact Riemannian manifolds are boundary distance rigid.

53C20 Global Riemannian geometry, including pinching
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