The boundary distance function and the Dirichlet-to-Neumann map
This paper consists in a detailed exposition of some main results on the following inverse problem: given a compact Riemannian manifold \((M, g)\), it is possible to determine \(g\) knowing the action of the distance function \(d_g\) on \(\partial M\times\partial M\). The authors announce the proof, in two dimensions, of a conjecture due to Michel, stating that on a simple manifold the function \(d_g\) determines \(g\) uniquely up to an isometry which is the identity map on the boundary. We recall that \((M, g)\) is called simple if \(\partial M\) is strictly convex and any two points of \(M\) are joined by a unique geodesic. The proof of the conjecture involves the scattering relation and the Dirichlet-Neumann map \(\Lambda_g\). In fact, \((\Lambda_g,\partial M)\) uniquely determines the conformal class of \(g\), when \(M\) is a compact surface. The authors prove that two simple metrics on a compact surface with the same scattering relation have the same Dirichlet-Neumann map. Finally, given two simple conformal metrics \(g\), \(\overline g= \rho g\) on the compact surface \(M\), the authors explain a procedure which allows to recover \(\rho\) from the distance function \(d_{\overline g}(x, y)\), \(x,y\in\partial M\).
scientific article; zbMATH DE number 2123929