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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.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
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