Siberian Federal University

Developing A.D. Aleksandrov’s ideas, the first author proposed the following approach to study of rigid- ity problems for the boundary of a C0-submanifold in a smooth Riemannian manifold. Let Y1 be a two-dimensional compact connected C0-submanifold with non-empty boundary in some smooth two- dimensional Riemannian manifold (X; g) without boundary. Let us consider the intrinsic metric (the infimum of the lengths of paths, connecting a pair of points".) of the interior Int Y1 of Y1, and extend it by continuity (operation lim ) to the boundary points of @Y1. In this paper the rigidity conditions are studied, i.e., when the constructed limiting metric defines @Y1 up to isometry of ambient space (X; g). We also consider the case dim Yj = dimX = n, n > 2