Rigidity and the distance between boundary points (Q806071)

Several rigidity results for Riemannian manifolds are proved. For example it is shown that any complete Riemannian metric without conjugate points on \({\mathbb{R}}^ n\) which is isometric to the Euclidean metric outside a compact set is isometric to the Euclidean metric. The results of the paper are related to the following ``boundary rigidity problem'' for a compact Riemannian manifold M with boundary H: Under which assumptions does the induced distance function d: \(H\times H\to {\mathbb{R}}\) determine the metric on M uniquely? This problem was studied earlier by R. Michel and M. Gromov.